How many questions are on the GRE Physics Subject Test and how long is it?

The computer-delivered test contains about 100 five-option multiple-choice questions with a time limit of 2 hours and 50 minutes (170 minutes). That works out to under two minutes per question on average, so pacing is a major part of the challenge.

How is the GRE Physics Subject Test scored?

ETS reports a single total scaled score from 200 to 990 (in 10-point increments) plus a percentile rank. For tests taken since September 2023, ETS also reports percent-correct subscores in Classical Mechanics, Electromagnetism, and Quantum Mechanics and Atomic Physics on a 0-100 scale.

What is a good GRE Physics score?

A good GRE Physics score is one with a high percentile rank, since the reported average total score is around 700 and the test is graded relative to other applicants. Competitive applicants to top graduate physics programs generally score well above the median, so aim for a high percentile rank rather than a fixed number.

How much does the GRE Physics Subject Test cost and how do I register?

The current published ETS fee is $150, which includes sending scores to up to four graduate institutions. You register online through your ETS account at ets.org and schedule a computer-delivered test during one of the administration windows offered each testing year. Always verify the current fee and test dates on the ETS site, since pricing and scheduling can change.

Do graduate programs still require the GRE Physics Subject Test?

Requirements vary by program and have shifted in recent years. Some physics graduate programs still require or recommend the Physics Subject Test, while many have made it optional or stopped using it. Always confirm each program's current admissions requirements directly before deciding to test.

What is the period of a simple pendulum of length $ L $ and negligible mass bob at a place where acceleration due to gravity is $ g $?

$ \frac{1}{2\pi}\sqrt{\frac{L}{g}} $

In a system of particles, the total momentum is conserved if:

Total kinetic energy is conserved

A planet orbits a star in an elliptical orbit. At which point is the planet's kinetic energy maximum?

At the midpoint of the major axis

A rigid body rotates about a fixed axis with a constant angular acceleration. Which of the following is true regarding the tangential acceleration of any point on the body?

It is directly proportional to the distance from the axis

It is inversely proportional to the distance from the axis

Two objects with masses $ m_1 $ and $ m_2 $ are connected by a light string over a frictionless pulley. If $ m_1 > m_2 $, what is the acceleration of the system? (Assume $ g $ is the acceleration due to gravity.)

$ g\frac{m_1 - m_2}{m_1 + m_2} $

$ g\frac{m_1 + m_2}{m_1 - m_2} $

A solid sphere and a hollow sphere, both of the same mass and radius, roll down an inclined plane. Which one reaches the bottom first?

Cannot be determined without knowing the angle of the incline

In a region of space, a uniform magnetic field passes perpendicularly through a rectangular loop of wire. If the magnetic field is steadily increasing, what is the direction of the induced current in the loop?

Counterclockwise, as viewed from above.

Clockwise, as viewed from above.

The direction of the current depends on the material of the wire.

What happens to the capacitance of a parallel-plate capacitor when a dielectric material is inserted between the plates?

The capacitance remains the same.

What is the magnetic field at the center of a circular loop of wire carrying a steady current?

Directed perpendicular to the plane of the loop

Directed parallel to the plane of the loop

Dependent on the radius of the loop

Which of the following best describes the principle of superposition in the context of electric fields?

The net electric field is the vector sum of all individual fields.

Electric fields can cancel each other out.

Electric fields always repel each other.

The strongest electric field dominates over weaker ones.

What is the self-inductance of a solenoid directly proportional to?

The square of the number of turns

What is the main effect of increasing the frequency of an alternating current in a purely resistive circuit?

There is no effect on the current or resistance.

Which of the following is true for the electric field inside a conductor in electrostatic equilibrium?

It is zero throughout the conductor.

It is maximum just inside the surface.

It is uniform throughout the conductor.

It varies inversely with the distance from the surface.

Which of the following is not a characteristic of electromagnetic waves?

They require a medium to travel.

They travel at the speed of light in a vacuum.

They consist of oscillating electric and magnetic fields.

What is the force between two parallel current-carrying wires when the currents are in opposite directions?

Variable, depending on the distance between the wires

A thin converging lens has a focal length of 15 cm. An object is placed 10 cm from the lens. Where is the image formed?

30 cm on the same side as the object

30 cm on the opposite side of the lens

A ray of light in air strikes a glass surface at an angle less than the critical angle. What happens to the ray?

It undergoes partial reflection and partial transmission

A thermally isolated system consisting of an ideal gas undergoes an adiabatic free expansion. Which of the following statements is true regarding this process?

The temperature of the gas remains constant.

The temperature of the gas increases.

The temperature of the gas decreases.

The pressure of the gas increases.

Which of the following best describes the Maxwell-Boltzmann distribution in statistical mechanics?

It is a distribution of velocities among particles in an ideal gas.

It is the distribution of energy states in a black body radiator.

It describes the distribution of particles over different quantum states in a Bose-Einstein condensate.

It represents the distribution of magnetic spin orientations in a paramagnetic material.

What is the primary implication of the Third Law of Thermodynamics?

The entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.

Energy cannot be created or destroyed.

The entropy of the universe is constantly increasing.

Heat cannot be converted entirely into work without some other change occurring at the same time.

In a reversible process, how does the entropy change in the universe?

It depends on the specific process.

What does the Stefan-Boltzmann law relate to in thermodynamics?

The total energy radiated per unit surface area of a black body.

The rate of heat transfer through conduction.

The pressure exerted by an ideal gas.

The change in entropy with temperature.

What is the significance of the Boltzmann constant in statistical mechanics?

It defines the relationship between temperature and energy.

It represents the maximum efficiency of a heat engine.

It indicates the strength of gravitational attraction between particles.

It measures the intensity of electric field due to a charged particle.

Your FREE GRE Physics Subject Test Practice Test 2026 – 160+ Q&A

Q: Is the GRE Physics Subject Test pass/fail?

No. It is an admissions test, not a pass/fail certification. Your total score is reported on a scaled range of 200 to 990 in 10-point increments, along with a percentile rank that shows how you performed relative to other test takers.

Q: What content areas does the GRE Physics Subject Test cover?

ETS distributes the test across nine areas: Classical Mechanics (20%), Electromagnetism (18%), Quantum Mechanics (12%), Atomic Physics (10%), Thermodynamics and Statistical Mechanics (10%), Optics and Wave Phenomena (9%), Specialized Topics (9%), Special Relativity (6%), and Laboratory Methods (6%).

Realistic GRE Physics Subject Test practice questions across every official ETS content area, with instant scoring and answer explanations.

Master questions to boost your scoreMaster questions to raise your exam readiness score

How ready are you?

GRE Physics Practice Questions

A particle moves in a circle of radius $r$ with constant speed $v$ . What is the magnitude of the acceleration of the particle?
- $\frac{v}{r}$
- $\frac{v^2}{r}$
- $vr$
- $\frac{r}{v^2}$
Correct answer: $\frac{v^2}{r}$
Correct answer: $\frac{v^2}{r}$ . Explanation: The acceleration in uniform circular motion is given by $\frac{v^2}{r}$ , directed towards the center of the circle.
A block of mass $m$ is on a frictionless inclined plane of angle $\theta$ . What is the acceleration of the block down the slope?
- $g\sin\theta$
- $g\cos\theta$
- $\frac{g}{\sin\theta}$
- $\frac{g}{\cos\theta}$
Correct answer: $g\sin\theta$
Correct answer: $g\sin\theta$ . Explanation: The component of gravitational force down the slope is $mg\sin\theta$ , leading to acceleration $g\sin\theta$ .
What is the period of a simple pendulum of length $L$ and negligible mass bob at a place where acceleration due to gravity is $g$ ?
- $2\pi\sqrt{\frac{L}{g}}$
- $\pi\sqrt{\frac{L}{g}}$
- $2\pi\sqrt{\frac{g}{L}}$
- $\frac{1}{2\pi}\sqrt{\frac{L}{g}}$
Correct answer: $2\pi\sqrt{\frac{L}{g}}$
Correct answer: $2\pi\sqrt{\frac{L}{g}}$ . Explanation: The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$ .
In a system of particles, the total momentum is conserved if:
- There are no external forces
- There is no net external force
- Total kinetic energy is conserved
- The system is in equilibrium
Correct answer: There is no net external force
Correct answer: There is no net external force. Explanation: Momentum conservation occurs in the absence of a net external force.
A planet orbits a star in an elliptical orbit. At which point is the planet's kinetic energy maximum?
- At the aphelion
- At the perihelion
- At any point on the orbit
- At the midpoint of the major axis
Correct answer: At the perihelion
Correct answer: At the perihelion. Explanation: The kinetic energy is maximum at the perihelion due to the highest orbital speed.
A rigid body rotates about a fixed axis with a constant angular acceleration. Which of the following is true regarding the tangential acceleration of any point on the body?
- It is zero
- It is constant for all points
- It is directly proportional to the distance from the axis
- It is inversely proportional to the distance from the axis
Correct answer: It is directly proportional to the distance from the axis
Correct answer: It is directly proportional to the distance from the axis. Explanation: Tangential acceleration is proportional to the radial distance from the axis of rotation.
A ball is thrown vertically upward with velocity $v$ . What is the time taken to reach its highest point?
- $\frac{v}{g}$
- $2\frac{v}{g}$
- $\frac{v^2}{g}$
- $\sqrt{\frac{2v}{g}}$
Correct answer: $\frac{v}{g}$
Correct answer: $\frac{v}{g}$ . Explanation: The time to reach the highest point is given by the initial velocity divided by acceleration due to gravity.
What is the angular frequency of a mass-spring system with mass $m$ and spring constant $k$ ?
- $\sqrt{\frac{k}{m}}$
- $\sqrt{\frac{m}{k}}$
- $\frac{k}{m}$
- $\frac{m}{k}$
Correct answer: $\sqrt{\frac{k}{m}}$
Correct answer: $\sqrt{\frac{k}{m}}$ . Explanation: The angular frequency of a mass-spring system is $\omega = \sqrt{\frac{k}{m}}$ .
The work done in slowly lifting an object to a height $h$ against gravity is equal to:
- $mgh$
- $\frac{1}{2}mgh$
- $mgh^2$
- Zero
Correct answer: $mgh$
Correct answer: $mgh$ . Explanation: The work done against gravity is equal to the gravitational potential energy, which is $mgh$ .
A particle of mass $m$ is moving in a circular path of radius $r$ with a constant speed $v$ . What is the angular momentum of the particle about the center of the circle?
- $mvr$
- $\frac{mv^2}{r}$
- $\frac{mv}{r}$
- $mvr^2$
Correct answer: $mvr$
Correct answer: $mvr$ . Explanation: The angular momentum $L$ is given by $L = mvr$ for a particle in uniform circular motion.
Two objects with masses $m_1$ and $m_2$ are connected by a light string over a frictionless pulley. If $m_1 > m_2$ , what is the acceleration of the system? (Assume $g$ is the acceleration due to gravity.)
- $g\frac{m_1 - m_2}{m_1 + m_2}$
- $g\frac{m_1 + m_2}{m_1 - m_2}$
- $g\frac{m_2}{m_1}$
- $g\frac{m_1}{m_2}$
Correct answer: $g\frac{m_1 - m_2}{m_1 + m_2}$
Correct answer: $g\frac{m_1 - m_2}{m_1 + m_2}$ . Explanation: The acceleration of the system in an Atwood machine is given by $a = g\frac{m_1 - m_2}{m_1 + m_2}$ .
A solid sphere and a hollow sphere, both of the same mass and radius, roll down an inclined plane. Which one reaches the bottom first?
- The solid sphere
- The hollow sphere
- Both at the same time
- Cannot be determined without knowing the angle of the incline
Correct answer: The solid sphere
Correct answer: The solid sphere. Explanation: The solid sphere has a smaller moment of inertia and therefore a larger acceleration, reaching the bottom first.
In a region of space, a uniform magnetic field passes perpendicularly through a rectangular loop of wire. If the magnetic field is steadily increasing, what is the direction of the induced current in the loop?
- Clockwise, as viewed from above.
- Counterclockwise, as viewed from above.
- No current is induced.
- The direction of the current depends on the material of the wire.
Correct answer: Counterclockwise, as viewed from above.
Correct answer: Counterclockwise, as viewed from above. Explanation: According to Lenz's Law, the induced current will flow in a direction to oppose the change in magnetic flux. Since the magnetic field is increasing, the induced current will be counterclockwise to create a magnetic field that opposes the increase.
What happens to the capacitance of a parallel-plate capacitor when a dielectric material is inserted between the plates?
- The capacitance decreases.
- The capacitance increases.
- The capacitance remains the same.
- The capacitance becomes zero.
Correct answer: The capacitance increases.
Correct answer: The capacitance increases. Explanation: Inserting a dielectric material between the plates of a parallel-plate capacitor increases its capacitance. This is because the dielectric reduces the electric field within the capacitor, allowing more charge to be stored for the same applied voltage.
What is the magnetic field at the center of a circular loop of wire carrying a steady current?
- Zero
- Directed perpendicular to the plane of the loop
- Directed parallel to the plane of the loop
- Dependent on the radius of the loop
Correct answer: Directed perpendicular to the plane of the loop
Correct answer: Directed perpendicular to the plane of the loop. Explanation: The magnetic field created by a current in a circular loop is perpendicular to the plane of the loop at its center, as determined by the right-hand rule.
How does the frequency of light change as it passes from air into water?
- It increases.
- It decreases.
- It remains the same.
- It becomes zero.
Correct answer: It remains the same.
Correct answer: It remains the same. Explanation: The frequency of light does not change when it passes from one medium to another. What changes is its wavelength and speed, but the frequency remains constant.
Which of the following best describes the principle of superposition in the context of electric fields?
- Electric fields can cancel each other out.
- The net electric field is the vector sum of all individual fields.
- Electric fields always repel each other.
- The strongest electric field dominates over weaker ones.
Correct answer: The net electric field is the vector sum of all individual fields.
Correct answer: The net electric field is the vector sum of all individual fields. Explanation: The principle of superposition states that the resultant electric field at a point is the vector sum of all individual electric fields acting at that point.
What is the SI unit of magnetic flux?
- Tesla
- Weber
- Henry
- Ampere
Correct answer: Weber
Correct answer: Weber. Explanation: The SI unit of magnetic flux is the Weber (Wb). It is defined as the magnetic flux that, linking a circuit of one turn, produces in it an electromotive force of one volt as it is reduced to zero at a uniform rate in one second.
In an AC circuit containing only a capacitor, what is the phase difference between the current and the voltage across the capacitor?
- 0 degrees
- 45 degrees
- 90 degrees
- 180 degrees
Correct answer: 90 degrees
Correct answer: 90 degrees. Explanation: In a purely capacitive AC circuit, the current leads the voltage by 90 degrees. This is because the current reaches its maximum value a quarter cycle before the voltage does.
What is the self-inductance of a solenoid directly proportional to?
- The square of the number of turns
- The resistance of the wire
- The current flowing through it
- The voltage across its ends
Correct answer: The square of the number of turns
Correct answer: The square of the number of turns. Explanation: The self-inductance of a solenoid is directly proportional to the square of the number of turns of the wire. It also depends on the cross-sectional area of the solenoid and the permeability of the core material.
What is the main effect of increasing the frequency of an alternating current in a purely resistive circuit?
- The resistance increases.
- The resistance decreases.
- The current increases.
- There is no effect on the current or resistance.
Correct answer: There is no effect on the current or resistance.
Correct answer: There is no effect on the current or resistance. Explanation: In a purely resistive circuit, the resistance remains constant regardless of the frequency of the alternating current. The current is determined by the voltage and the resistance according to Ohm's law.
Which of the following is true for the electric field inside a conductor in electrostatic equilibrium?
- It is maximum just inside the surface.
- It is zero throughout the conductor.
- It is uniform throughout the conductor.
- It varies inversely with the distance from the surface.
Correct answer: It is zero throughout the conductor.
Correct answer: It is zero throughout the conductor. Explanation: In a conductor in electrostatic equilibrium, the electric field inside the conductor is zero. This is because the free charges within the conductor rearrange themselves to cancel out any external field.
Which of the following is not a characteristic of electromagnetic waves?
- They require a medium to travel.
- They travel at the speed of light in a vacuum.
- They are transverse waves.
- They consist of oscillating electric and magnetic fields.
Correct answer: They require a medium to travel.
Correct answer: They require a medium to travel. Explanation: Electromagnetic waves do not require a medium to travel; they can propagate through a vacuum. This is unlike mechanical waves, which require a medium.
What is the force between two parallel current-carrying wires when the currents are in opposite directions?
- Attractive
- Repulsive
- Zero
- Variable, depending on the distance between the wires
Correct answer: Repulsive
Correct answer: Repulsive. Explanation: When two parallel wires carry currents in opposite directions, they exert a repulsive force on each other. This is due to the interaction of the magnetic fields produced by the currents in each wire.
A thin converging lens has a focal length of 15 cm. An object is placed 10 cm from the lens. Where is the image formed?
- 30 cm on the same side as the object
- 30 cm on the opposite side of the lens
- At infinity
- No image is formed
Correct answer: 30 cm on the same side as the object
Correct answer: 30 cm on the same side as the object. Explanation: Use the lens formula $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ to find the image distance.
Which phenomenon is not explained by the wave theory of light?
- Interference
- Diffraction
- Photoelectric effect
- Polarization
Correct answer: Photoelectric effect
Correct answer: Photoelectric effect. Explanation: The photoelectric effect is explained by the particle theory of light (photon theory), not by the wave theory.
What happens to the speed of light as it enters a medium with a higher refractive index?
- It increases
- It decreases
- It remains constant
- It becomes zero
Correct answer: It decreases
Correct answer: It decreases. Explanation: The speed of light decreases when it enters a medium with a higher refractive index.
In a double-slit experiment, if the slit separation is halved and the wavelength of light used is doubled, what happens to the fringe spacing?
- It is halved
- It remains the same
- It is doubled
- It is quadrupled
Correct answer: It is doubled
Correct answer: It is doubled. Explanation: Fringe spacing is directly proportional to the wavelength and inversely proportional to slit separation.
What is the Brewster's angle for a medium with a refractive index of 1.5?
- $\tan^{-1}(1/1.5)$
- $\tan^{-1}(1.5)$
- $\sin^{-1}(1/1.5)$
- $\cos^{-1}(1/1.5)$
Correct answer: $\tan^{-1}(1.5)$
Correct answer: $\tan^{-1}(1.5)$ . Explanation: Brewster's angle is given by $\tan^{-1}(n)$ where $n$ is the refractive index.
Which color of light has the shortest wavelength when passing through a prism?
- Red
- Green
- Blue
- Violet
Correct answer: Violet
Correct answer: Violet. Explanation: Violet light has the shortest wavelength among visible colors in the spectrum.
A beam of unpolarized light is incident on a polarizer. What fraction of the light's intensity is transmitted through the polarizer?
- $\frac{1}{4}$
- $\frac{1}{2}$
- $\frac{3}{4}$
- $1$
Correct answer: $\frac{1}{2}$
Correct answer: $\frac{1}{2}$ . Explanation: A polarizer transmits half the intensity of an incident unpolarized light beam.
In an optical fiber, light is confined within the core by the principle of:
- Refraction
- Diffraction
- Total internal reflection
- Scattering
Correct answer: Total internal reflection
Correct answer: Total internal reflection. Explanation: Optical fibers use total internal reflection to confine light within the core.
A ray of light in air strikes a glass surface at an angle less than the critical angle. What happens to the ray?
- It is completely reflected
- It is completely transmitted
- It undergoes partial reflection and partial transmission
- It is absorbed by the glass
Correct answer: It undergoes partial reflection and partial transmission
Correct answer: It undergoes partial reflection and partial transmission. Explanation: At angles less than the critical angle, light undergoes both reflection and transmission at the interface.
A thermally isolated system consisting of an ideal gas undergoes an adiabatic free expansion. Which of the following statements is true regarding this process?
- The temperature of the gas increases.
- The temperature of the gas decreases.
- The temperature of the gas remains constant.
- The pressure of the gas increases.
Correct answer: The temperature of the gas remains constant.
Correct answer: The temperature of the gas remains constant. Explanation: In an adiabatic free expansion of an ideal gas, there is no external work done on or by the system, and no heat exchange with the surroundings. Thus, for an ideal gas, the temperature remains constant.
Which of the following best describes the Maxwell-Boltzmann distribution in statistical mechanics?
- It is a distribution of velocities among particles in an ideal gas.
- It is the distribution of energy states in a black body radiator.
- It describes the distribution of particles over different quantum states in a Bose-Einstein condensate.
- It represents the distribution of magnetic spin orientations in a paramagnetic material.
Correct answer: It is a distribution of velocities among particles in an ideal gas.
Correct answer: It is a distribution of velocities among particles in an ideal gas. Explanation: The Maxwell-Boltzmann distribution describes the distribution of speeds (or equivalently, kinetic energies) of particles in an ideal gas. This statistical distribution is based on classical mechanics.
If the internal energy of an ideal gas depends only on its temperature, which of the following processes will result in no change in internal energy?
- Isothermal expansion
- Isobaric compression
- Isochoric heating
- Adiabatic expansion
Correct answer: Isothermal expansion
Correct answer: Isothermal expansion. Explanation: In an isothermal process, the temperature of the system remains constant. Since the internal energy of an ideal gas depends only on temperature, an isothermal process will result in no change in internal energy.
Which principle is best illustrated by the operation of a heat pump?
- Conservation of energy
- Second law of thermodynamics
- Pascal's principle
- Bernoulli's principle
Correct answer: Second law of thermodynamics
Correct answer: Second law of thermodynamics. Explanation: A heat pump demonstrates the second law of thermodynamics, which states that heat energy cannot spontaneously flow from a colder location to a hotter area without external work being done on the system.
What is the primary implication of the Third Law of Thermodynamics?
- The entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.
- Energy cannot be created or destroyed.
- The entropy of the universe is constantly increasing.
- Heat cannot be converted entirely into work without some other change occurring at the same time.
Correct answer: The entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.
Correct answer: The entropy of a perfect crystal approaches zero as the temperature approaches absolute zero. Explanation: The Third Law of Thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero. For a perfectly ordered crystal at 0 K, the entropy is typically zero.
In a reversible process, how does the entropy change in the universe?
- It increases.
- It decreases.
- It remains constant.
- It depends on the specific process.
Correct answer: It remains constant.
Correct answer: It remains constant. Explanation: In a reversible process, the total entropy change in the universe (system plus surroundings) is zero. This is because reversible processes are idealized scenarios where the system is always in thermodynamic equilibrium.
What does the Stefan-Boltzmann law relate to in thermodynamics?
- The rate of heat transfer through conduction.
- The pressure exerted by an ideal gas.
- The total energy radiated per unit surface area of a black body.
- The change in entropy with temperature.
Correct answer: The total energy radiated per unit surface area of a black body.
Correct answer: The total energy radiated per unit surface area of a black body. Explanation: The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths is directly proportional to the fourth power of the black body's temperature.
What is the significance of the Boltzmann constant in statistical mechanics?
- It defines the relationship between temperature and energy.
- It represents the maximum efficiency of a heat engine.
- It indicates the strength of gravitational attraction between particles.
- It measures the intensity of electric field due to a charged particle.
Correct answer: It defines the relationship between temperature and energy.
Correct answer: It defines the relationship between temperature and energy. Explanation: The Boltzmann constant (k) plays a crucial role in statistical mechanics, linking the average kinetic energy of particles in a gas with the temperature of the gas. It is a bridge between macroscopic and microscopic physics.
A Carnot engine operates between two reservoirs at temperatures $T_1$ and $T_2$ ( $T_1 > T_2$ ). If the efficiency of the engine is 40\%, what is the ratio $\frac{T_1}{T_2}$ ?
- $\frac{5}{3}$
- $\frac{4}{3}$
- $\frac{7}{3}$
- $\frac{3}{2}$
Correct answer: $\frac{5}{3}$
Correct answer: $\frac{5}{3}$ . Explanation: The efficiency of a Carnot engine is given by $1 - \frac{T_2}{T_1}$ . Solving $0.4 = 1 - \frac{T_2}{T_1}$ gives $\frac{T_1}{T_2} = \frac{5}{3}$ .
In a thermodynamic process, a gas expands from volume $V_1$ to $V_2$ and absorbs $Q$ amount of heat. If the work done by the gas is $W$ and the change in internal energy is $\Delta U = 0$ , what is the relationship between $Q$ and $W$ ?
- $Q = W$
- $Q = 2W$
- $Q = \frac{W}{2}$
- $Q = -W$
Correct answer: $Q = W$
Correct answer: $Q = W$ . Explanation: By the first law of thermodynamics, $\Delta U = Q - W$ . If $\Delta U = 0$ , then $Q = W$ .
A particle in a one-dimensional box of width $L$ is in a state described by the wave function $\psi(x) = A \sin(\frac{2\pi x}{L})$ . What is the energy of the particle?
- $\frac{h^2}{8mL^2}$
- $\frac{h^2}{4mL^2}$
- $\frac{h^2}{2mL^2}$
- $\frac{h^2}{mL^2}$
Correct answer: $\frac{h^2}{2mL^2}$
Correct answer: $\frac{h^2}{2mL^2}$ . Explanation: The energy levels in a one-dimensional box are given by $E_n = \frac{n^2h^2}{8mL^2}$ . For $n = 2$ , it simplifies to $\frac{h^2}{2mL^2}$ .
Which of the following represents the correct commutation relation between position ( $\hat{x}$ ) and momentum ( $\hat{p}$ ) operators?
- $[\hat{x}, \hat{p}] = i\hbar$
- $[\hat{x}, \hat{p}] = \hbar$
- $[\hat{x}, \hat{p}] = -i\hbar$
- $[\hat{x}, \hat{p}] = 0$
Correct answer: $[\hat{x}, \hat{p}] = i\hbar$
Correct answer: $[\hat{x}, \hat{p}] = i\hbar$ . Explanation: The commutator $[\hat{x}, \hat{p}]$ is defined as $\hat{x}\hat{p} - \hat{p}\hat{x}$ and is equal to $i\hbar$ in quantum mechanics.
Which principle states that no two fermions can occupy the same quantum state?
- Heisenberg's uncertainty principle
- Pauli exclusion principle
- Hund's rule
- Aufbau principle
Correct answer: Pauli exclusion principle
Correct answer: Pauli exclusion principle. Explanation: The Pauli exclusion principle states that no two fermions can occupy the same quantum state simultaneously.
In quantum mechanics, what does the operator $\hat{H}$ represent?
- Momentum
- Position
- Hamiltonian
- Spin
Correct answer: Hamiltonian
Correct answer: Hamiltonian. Explanation: The operator $\hat{H}$ in quantum mechanics represents the Hamiltonian, which corresponds to the total energy of the system.
If a quantum system is described by the normalized wave function $\psi(x)$ , what is the probability of finding the particle in the interval $a \leq x \leq b$ ?
- $\int_a^b \psi(x) dx$
- $\int_a^b |\psi(x)|^2 dx$
- $|\int_a^b \psi(x) dx|^2$
- $\int_a^b |\psi(x)| dx$
Correct answer: $\int_a^b |\psi(x)|^2 dx$
Correct answer: $\int_a^b |\psi(x)|^2 dx$ . Explanation: The probability of finding a particle in a given interval is the integral of the square of the magnitude of the wave function over that interval.
What is the spin quantum number for an electron?
- $1/2$
- $1$
- $3/2$
- $2$
Correct answer: $1/2$
Correct answer: $1/2$ . Explanation: Electrons are fermions with a spin quantum number of $1/2$ .
In the context of quantum mechanics, what does the term 'orthonormality' refer to?
- The condition where operators commute
- The conservation of probability
- The property of wave functions being orthogonal and normalized
- The linear superposition of states
Correct answer: The property of wave functions being orthogonal and normalized
Correct answer: The property of wave functions being orthogonal and normalized. Explanation: Orthonormality in quantum mechanics refers to the property where wave functions are orthogonal to each other and each function is normalized.
In the context of quantum mechanics, which principle states that it is impossible to simultaneously know both the exact momentum and position of a particle?
- Heisenberg Uncertainty Principle
- Pauli Exclusion Principle
- Bohr's Correspondence Principle
- Planck's Quantization Principle
Correct answer: Heisenberg Uncertainty Principle
Correct answer: Heisenberg Uncertainty Principle. Explanation: The Heisenberg Uncertainty Principle asserts that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.
Which phenomenon demonstrates the wave-particle duality of electrons?
- Rutherford scattering
- Electron diffraction
- Photoelectric effect
- Compton scattering
Correct answer: Electron diffraction
Correct answer: Electron diffraction. Explanation: Electron diffraction demonstrates the wave-like properties of electrons, showing wave-particle duality. It occurs when electrons display wave-like behavior, such as interference patterns.
In a hydrogen atom, which electronic transition corresponds to the emission of the longest wavelength photon?
- N=2 to n=1
- N=3 to n=2
- N=3 to n=1
- N=2 to n=3
Correct answer: N=3 to n=2
Correct answer: n=3 to n=2. Explanation: The longest wavelength photon corresponds to the smallest energy transition. The transition from n=3 to n=2 in a hydrogen atom has the smallest energy difference among the given options.
In atomic physics, what does the fine structure of spectral lines primarily result from?
- Doppler effect
- Zeeman effect
- Spin-orbit coupling
- Hyperfine splitting
Correct answer: Spin-orbit coupling
Correct answer: Spin-orbit coupling. Explanation: The fine structure of spectral lines in atomic physics is primarily due to spin-orbit coupling, which is the interaction of the electron's spin with its orbital motion around the nucleus.
Which type of quantum numbers are used to describe the state of an electron in an atom?
- Principal, azimuthal, magnetic, and spin
- Kinetic, potential, total energy, and angular momentum
- Mass, charge, velocity, and spin
- Frequency, wavelength, amplitude, and phase
Correct answer: Principal, azimuthal, magnetic, and spin
Correct answer: Principal, azimuthal, magnetic, and spin. Explanation: The state of an electron in an atom is described by four quantum numbers: principal (n), azimuthal (l), magnetic (m), and spin (s).
What happens to the energy levels of an atom in a strong magnetic field (Zeeman effect)?
- They remain unchanged.
- They split into multiple closely spaced levels.
- They converge into a single level.
- They become continuous.
Correct answer: They split into multiple closely spaced levels.
Correct answer: They split into multiple closely spaced levels. Explanation: In the presence of a strong magnetic field (Zeeman effect), the energy levels of an atom split into multiple closely spaced levels. This is due to the interaction of the magnetic field with the magnetic moments of electrons.
Which principle explains the stability of the electron in an atom, preventing it from falling into the nucleus?
- Pauli Exclusion Principle
- Heisenberg Uncertainty Principle
- Coulomb's Law
- Conservation of energy
Correct answer: Heisenberg Uncertainty Principle
Correct answer: Heisenberg Uncertainty Principle. Explanation: The Heisenberg Uncertainty Principle, which states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision, explains the stability of the electron in an atom.
What is the primary cause of hyperfine splitting in atomic spectra?
- Interaction between the nuclear spin and the electron cloud.
- The relativistic effect of high electron velocity.
- The effect of external electric fields on atomic energy levels.
- Collisions between atoms.
Correct answer: Interaction between the nuclear spin and the electron cloud.
Correct answer: Interaction between the nuclear spin and the electron cloud. Explanation: Hyperfine splitting in atomic spectra is primarily caused by the interaction between the nuclear spin and the electron cloud surrounding the nucleus.
What is the primary effect observed in the Stern-Gerlach experiment?
- The splitting of an atomic beam into discrete components due to the interaction of atomic magnetic moments with a magnetic field.
- The deflection of alpha particles when they pass near a nucleus.
- The interference pattern formed by electrons passing through a double slit.
- The emission of photons when electrons transition between energy levels.
Correct answer: The splitting of an atomic beam into discrete components due to the interaction of atomic magnetic moments with a magnetic field.
Correct answer: The splitting of an atomic beam into discrete components due to the interaction of atomic magnetic moments with a magnetic field. Explanation: The Stern-Gerlach experiment demonstrates the quantization of angular momentum and magnetic moments in atoms. In this experiment, an atomic beam is split into discrete components when it passes through a non-uniform magnetic field, illustrating the quantum nature of atomic magnetic moments.
What happens to the mass of a particle as it approaches the speed of light?
- It remains constant.
- It approaches zero.
- It increases indefinitely.
- It decreases to half its rest mass.
Correct answer: It increases indefinitely.
Correct answer: It increases indefinitely. Explanation: According to special relativity, as a particle approaches the speed of light, its relativistic mass increases indefinitely due to the increase in energy required to accelerate it.
What is time dilation in special relativity?
- The decrease in the rate of time as observed in a moving frame of reference.
- The increase in the rate of time as observed in a moving frame of reference.
- The unchanging rate of time regardless of the frame of reference.
- The relative perception of time in different gravitational fields.
Correct answer: The decrease in the rate of time as observed in a moving frame of reference.
Correct answer: The decrease in the rate of time as observed in a moving frame of reference. Explanation: Time dilation is a phenomenon in special relativity where the observed rate of time is slower in a moving frame of reference compared to a stationary frame of reference.
Which of the following best describes the principle of relativity?
- The laws of physics are the same in all inertial frames.
- The speed of light is constant in all frames of reference.
- The laws of physics vary with gravitational fields.
- The speed of light changes with the observer's motion.
Correct answer: The laws of physics are the same in all inertial frames.
Correct answer: The laws of physics are the same in all inertial frames. Explanation: The principle of relativity, as formulated by Einstein, states that the laws of physics are invariant (i.e., identical) in all inertial frames of reference.
How does the length of an object change as it approaches the speed of light?
- It remains constant.
- It increases.
- It decreases along the direction of motion.
- It decreases perpendicular to the direction of motion.
Correct answer: It decreases along the direction of motion.
Correct answer: It decreases along the direction of motion. Explanation: Length contraction in special relativity states that objects physically contract along the direction of motion as they approach the speed of light.
What is the relativistic effect observed when an object is moving at a significant fraction of the speed of light?
- Time dilation and mass reduction.
- Time dilation and length contraction.
- Speed increase and mass increase.
- Speed increase and time acceleration.
Correct answer: Time dilation and length contraction.
Correct answer: Time dilation and length contraction. Explanation: Special relativity predicts that time will appear to move slower (time dilation) and lengths will appear shorter (length contraction) in an object moving at a significant fraction of the speed of light.
In the twin paradox of special relativity, what is the primary reason for the age difference observed between the twins?
- The gravitational time dilation effect.
- The acceleration and deceleration of the traveling twin.
- The constant speed of light in all reference frames.
- The difference in mass between the twins.
Correct answer: The acceleration and deceleration of the traveling twin.
Correct answer: The acceleration and deceleration of the traveling twin. Explanation: The age difference in the twin paradox arises because the traveling twin undergoes acceleration and deceleration, breaking the symmetry of the situation and leading to a real difference in ages due to time dilation.
Which of the following scenarios accurately illustrates length contraction?
- A spaceship traveling at near-light speed appears shorter to a stationary observer.
- A stationary object appears longer to a moving observer.
- An object's length increases as its speed decreases.
- Length contraction occurs only at speeds close to the speed of sound.
Correct answer: A spaceship traveling at near-light speed appears shorter to a stationary observer.
Correct answer: A spaceship traveling at near-light speed appears shorter to a stationary observer. spaceship traveling at near-light speed appears shorter to a stationary observer. Explanation: Length contraction is a phenomenon predicted by special relativity, where objects moving at a significant fraction of the speed of light will appear contracted in the direction of motion to a stationary observer.
How does the relativistic Doppler effect differ from the classical Doppler effect?
- It only applies to light waves, not sound waves.
- It includes both the velocity of the source and the observer.
- It accounts for the time dilation effect.
- It occurs only when the source and observer are moving towards each other.
Correct answer: It accounts for the time dilation effect.
Correct answer: It accounts for the time dilation effect. Explanation: The relativistic Doppler effect differs from the classical Doppler effect in that it takes into account the time dilation effect due to the relative motion at significant fractions of the speed of light.
What is the significance of the invariant interval in special relativity?
- It is the distance that remains constant between two events in space-time.
- It refers to the constant speed of light in vacuum.
- It is the period during which the laws of physics are invariant.
- It indicates the energy required to accelerate an object to the speed of light.
Correct answer: It is the distance that remains constant between two events in space-time.
Correct answer: It is the distance that remains constant between two events in space-time. Explanation: The invariant interval in special relativity is a quantity that remains the same for all observers, regardless of their relative motion. It is a fundamental aspect of the space-time continuum.
In the context of special relativity, what happens to the observed frequency of light from a source moving away from the observer at a significant fraction of the speed of light?
- The frequency increases (blueshift).
- The frequency remains unchanged.
- The frequency decreases (redshift).
- The frequency alternates between increasing and decreasing.
Correct answer: The frequency decreases (redshift).
Correct answer: The frequency decreases (redshift). Explanation: According to special relativity, if a light source is moving away from an observer at a significant fraction of the speed of light, the observed frequency of the light decreases, a phenomenon known as redshift. This is a consequence of the Doppler effect in relativistic contexts.
What is the effect of relativistic speeds on the observed color of a star moving rapidly towards the Earth?
- The star appears redder.
- The star appears bluer.
- There is no change in the observed color.
- The star appears greener.
Correct answer: The star appears bluer.
Correct answer: The star appears bluer. Explanation: As a star moves rapidly towards the Earth at relativistic speeds, the light it emits experiences a blueshift. This is due to the decrease in the wavelength of light, causing the star to appear bluer to an observer on Earth.
In special relativity, what happens to the kinetic energy of an object as its speed approaches the speed of light?
- It approaches a maximum finite value.
- It decreases to zero.
- It increases without bound.
- It remains constant.
Correct answer: It increases without bound.
Correct answer: It increases without bound. Explanation: As an object's speed approaches the speed of light, its relativistic kinetic energy increases without bound. This is due to the relativistic increase in mass and the fact that an infinite amount of energy would be required to reach the speed of light.
Which of the following best describes the concept of 'proper time' in special relativity?
- The time measured by an observer in motion relative to the event.
- The time interval measured in a reference frame where the event is at rest.
- The time dilation experienced by an observer moving at the speed of light.
- The time difference measured by two observers in different gravitational fields.
Correct answer: The time interval measured in a reference frame where the event is at rest.
Correct answer: The time interval measured in a reference frame where the event is at rest. Explanation: Proper time in special relativity is defined as the time interval measured by a clock that is at rest relative to the event being timed. It is the shortest time interval between two events in spacetime.
When using a lock-in amplifier in a signal detection setup, what is the primary advantage of this technique?
- It increases the signal strength.
- It allows for the measurement of phase.
- It reduces noise by filtering out signals at different frequencies.
- It converts AC signals to DC.
Correct answer: It reduces noise by filtering out signals at different frequencies.
Correct answer: It reduces noise by filtering out signals at different frequencies. Explanation: A lock-in amplifier is primarily used to reduce noise in measurements. It achieves this by filtering out all frequencies except for the one it is 'locked-in' to, thus significantly reducing noise.
In a cryogenic experiment, which material is typically used as a coolant to achieve temperatures close to absolute zero?
- Liquid nitrogen
- Liquid helium
- Dry ice
- Liquid oxygen
Correct answer: Liquid helium
Correct answer: Liquid helium. Explanation: Liquid helium is used in cryogenic experiments to reach temperatures close to absolute zero due to its very low boiling point of 4.2 K.
What is the main purpose of a Faraday cage in an experimental setup?
- To shield the experiment from external electric fields.
- To contain electromagnetic radiation within the cage.
- To enhance magnetic fields inside the cage.
- To isolate the experiment from thermal fluctuations.
Correct answer: To shield the experiment from external electric fields.
Correct answer: To shield the experiment from external electric fields. Explanation: A Faraday cage is used to shield its contents from external static electric fields. It does this by distributing charges on its exterior, thus canceling external fields.
In a laser cooling experiment, what is the primary mechanism for reducing the temperature of atoms?
- Adiabatic demagnetization
- Doppler cooling
- Joule-Thomson expansion
- Evaporative cooling
Correct answer: Doppler cooling
Correct answer: Doppler cooling. Explanation: Doppler cooling is the primary mechanism in laser cooling experiments. It involves the use of laser light to slow down atoms, effectively reducing their temperature.
In an optical tweezers setup, what is the fundamental principle that allows for the trapping and manipulation of small particles?
- Magnetic forces
- Electric field gradients
- Gravitational forces
- Radiation pressure
Correct answer: Radiation pressure
Correct answer: Radiation pressure. Explanation: Optical tweezers use the radiation pressure of light to trap and manipulate microscopic particles. This pressure is exerted by the momentum of photons.
In a Hall effect measurement setup, what property of a material is directly measured?
- Electrical resistivity
- Thermal conductivity
- Carrier density
- Magnetic susceptibility
Correct answer: Carrier density
Correct answer: Carrier density. Explanation: The Hall effect is used to measure the carrier density (number of charge carriers per unit volume) in a material.
What is the primary reason for using a cryostat in low-temperature physics experiments?
- To maintain a vacuum
- To provide thermal insulation
- To measure electrical resistance
- To apply a magnetic field
Correct answer: To provide thermal insulation
Correct answer: To provide thermal insulation. Explanation: A cryostat is used in low-temperature experiments primarily to provide thermal insulation, thereby maintaining the low temperatures necessary for such experiments.
In a time-of-flight (TOF) mass spectrometry experiment, what physical quantity is measured to determine the mass of ions?
- Velocity
- Time
- Electric charge
- Magnetic field strength
Correct answer: Time
Correct answer: Time. Explanation: In TOF mass spectrometry, the time it takes for ions to travel a known distance is measured. This time is used to calculate their mass-to-charge ratio.
In a Fourier-transform infrared (FTIR) spectroscopy experiment, what is the key advantage of this technique compared to dispersive IR spectroscopy?
- Higher resolution
- Faster data acquisition
- Better signal-to-noise ratio
- All of the above
Correct answer: All of the above
Correct answer: All of the above. Explanation: FTIR spectroscopy offers several advantages over dispersive IR, including higher resolution, faster data acquisition, and a better signal-to-noise ratio due to the Fellgett's advantage.
In the context of nuclear physics, what does the term "magic number" refer to?
- The number of protons or neutrons in a nucleus which gives rise to extra stability.
- The total energy required to disassemble a nucleus into its constituent protons and neutrons.
- The minimum energy required to remove a neutron from a nucleus.
- The energy released during a nuclear fission process.
Correct answer: The number of protons or neutrons in a nucleus which gives rise to extra stability.
Correct answer: The number of protons or neutrons in a nucleus which gives rise to extra stability. Explanation: In nuclear physics, "magic numbers" refer to the number of protons or neutrons in a nucleus that are arranged in complete shells within the atomic nucleus. Nuclei with magic numbers of protons or neutrons are more stable.
What is the primary characteristic of a particle in a "superfluid" state?
- It exhibits zero viscosity.
- It has a maximum entropy state.
- It displays increased electrical conductivity.
- It undergoes rapid thermal expansion.
Correct answer: It exhibits zero viscosity.
Correct answer: It exhibits zero viscosity. Explanation: Superfluidity is a phase of matter characterized by the complete absence of viscosity. Particles in a superfluid state can flow without losing kinetic energy.
What does the Chandrasekhar limit signify in astrophysics?
- The maximum mass of a stable white dwarf star.
- The minimum mass required for a star to initiate nuclear fusion.
- The distance at which a star's gravitational pull equals its radiation pressure.
- The temperature at which nuclear fusion stops in a star's core.
Correct answer: The maximum mass of a stable white dwarf star.
Correct answer: The maximum mass of a stable white dwarf star. Explanation: The Chandrasekhar limit is the maximum mass (approximately 1.4 solar masses) that a stable white dwarf star can have. Beyond this limit, the star would collapse into a neutron star or black hole.
What phenomenon does the "Casimir effect" demonstrate?
- The attraction or repulsion between two parallel, uncharged conducting plates.
- The increase in velocity of a fluid when it passes through a constricted section.
- The diffraction patterns caused by waves encountering an obstacle.
- The emission of electrons from a material when light shines on it.
Correct answer: The attraction or repulsion between two parallel, uncharged conducting plates.
Correct answer: The attraction or repulsion between two parallel, uncharged conducting plates. Explanation: The Casimir effect is a quantum mechanical phenomenon where two uncharged metallic plates in a vacuum attract or repel each other due to quantum fluctuations in the vacuum.
What is the primary outcome of the "Klein paradox" in quantum mechanics?
- It demonstrates the impossibility of faster-than-light travel.
- It shows that under certain conditions, potential barriers can become transparent to incoming particles.
- It describes the phenomenon where particles and antiparticles annihilate each other.
- It explains the stability of orbitals in an atom.
Correct answer: It shows that under certain conditions, potential barriers can become transparent to incoming particles.
Correct answer: It shows that under certain conditions, potential barriers can become transparent to incoming particles. Explanation: The Klein paradox is a counterintuitive phenomenon in quantum mechanics, where under certain conditions, a potential barrier can become transparent to incoming particles regardless of the barrier's height or width, a phenomenon known as tunneling.
In the context of condensed matter physics, what is "quantum Hall effect" associated with?
- The quantization of the Hall voltage in a two-dimensional electron gas.
- The increase in electrical resistance in conductors at low temperatures.
- The change in magnetic properties of materials at quantum-level temperatures.
- The superconducting state in high-temperature superconductors.
Correct answer: The quantization of the Hall voltage in a two-dimensional electron gas.
Correct answer: The quantization of the Hall voltage in a two-dimensional electron gas. Explanation: The quantum Hall effect refers to the quantization of the Hall voltage observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, leading to the formation of discrete energy levels.
What is the defining characteristic of a "Bose-Einstein condensate"?
- A state of matter formed by fermions at extremely low temperatures.
- A state of matter where particles exhibit integer spin and occupy the same ground state.
- A plasma state where ionization of gases occurs.
- A liquid crystal phase seen in certain organic materials.
Correct answer: A state of matter where particles exhibit integer spin and occupy the same ground state.
Correct answer: A state of matter where particles exhibit integer spin and occupy the same ground state. state of matter where particles exhibit integer spin and occupy the same ground state. Explanation: A Bose-Einstein condensate is a state of matter formed by bosons cooled to temperatures very close to absolute zero. Under such conditions, a large fraction of bosons occupy the same ground state, exhibiting macroscopic quantum phenomena.
In particle physics, what does "color confinement" refer to?
- The inability of individual colors (charges) of quarks to be isolated.
- The spectrum of visible light emitted by heated particles.
- The confinement of electrons in specific atomic orbitals.
- The trapping of photons in black holes.
Correct answer: The inability of individual colors (charges) of quarks to be isolated.
Correct answer: The inability of individual colors (charges) of quarks to be isolated. Explanation: Color confinement is a phenomenon in quantum chromodynamics QCD where quarks and gluons cannot be separated from their parent hadrons, meaning the "color charge" of quarks is never observed in isolation.
What principle does the "Aharonov-Bohm effect" illustrate in quantum mechanics?
- The effect of a magnetic field on the phase of a particle's wave function.
- The change in wavelength of light due to the motion of the source.
- The increase in mass of particles as they approach the speed of light.
- The interference pattern produced by electrons passing through two slits.
Correct answer: The effect of a magnetic field on the phase of a particle's wave function.
Correct answer: The effect of a magnetic field on the phase of a particle's wave function. Explanation: The Aharonov-Bohm effect demonstrates that a magnetic field can affect the phase of a particle's wave function in a region where the magnetic field is not present, indicating the fundamental nature of potential in quantum mechanics.
What property of a wave remains constant as it passes from one medium to another?
- Wavelength
- Frequency
- Speed
- Amplitude
Correct answer: Frequency
Correct answer: Frequency. Explanation: The frequency of a wave remains constant when transitioning between media.
In the rotating frame of the Earth, the Coriolis acceleration on a moving object is given by which expression, where omega is the planet's angular velocity vector and v is the velocity measured in the rotating frame?
- Minus 2 times the cross product of omega and v
- The cross product of v and the gravitational field g
- Minus 2 times omega times v, directed along the rotation axis
- Minus omega squared times the position vector r
Correct answer: Minus 2 times the cross product of omega and v
The Coriolis acceleration is minus 2 times the cross product of omega and v. It is a fictitious acceleration that appears only in a rotating reference frame, it is proportional to the object's speed and the rotation rate, and it is always perpendicular to both the rotation axis and the velocity. The expression minus omega squared r is the centrifugal acceleration, a separate inertial-frame correction.
A projectile is fired horizontally toward the north in the Northern Hemisphere. Ignoring air resistance, which way does the Coriolis effect deflect it relative to its direction of motion?
- To the left of its motion (toward the west)
- There is no horizontal deflection at all
- Straight up, opposing gravity
- To the right of its motion (toward the east)
Correct answer: To the right of its motion (toward the east)
The Coriolis effect deflects the projectile to the right of its motion (toward the east) in the Northern Hemisphere. Because the Coriolis acceleration is minus 2 omega cross v, applying the right-hand rule with the Earth's angular velocity pointing out of the North Pole gives a horizontal deflection to the right for any horizontal motion in the Northern Hemisphere. In the Southern Hemisphere the sense reverses, deflecting moving objects to the left.
A thin uniform rod of mass M and length L has a moment of inertia of (1/12) M L squared about an axis through its center perpendicular to the rod. Using the parallel axis theorem, what is its moment of inertia about a parallel axis through one end?
- (1/6) M L squared
- (5/12) M L squared
- (1/3) M L squared
- (1/4) M L squared
Correct answer: (1/3) M L squared
The moment of inertia about the end is (1/3) M L squared. The parallel axis theorem states I equals I_cm plus M d squared, where d is the distance from the center of mass to the new axis; here d equals L/2, so I equals (1/12) M L squared plus M (L/2) squared equals (1/12 plus 1/4) M L squared equals (1/3) M L squared. The value (1/12) M L squared applies only to the central axis.
The moment of inertia of a uniform solid sphere of mass M and radius R about an axis through its center is (2/5) M R squared. What is its moment of inertia about an axis tangent to its surface?
- (3/5) M R squared
- (2/5) M R squared
- (7/5) M R squared
- (12/5) M R squared
Correct answer: (7/5) M R squared
The moment of inertia about a tangent axis is (7/5) M R squared. By the parallel axis theorem, I equals (2/5) M R squared plus M R squared, since the tangent axis lies a distance R from the parallel central axis, giving (2/5 plus 1) M R squared equals (7/5) M R squared. This is a standard application of the theorem to the solid-sphere result of (2/5) M R squared.
A net force does 60 joules of work on a 3.0 kilogram object that starts from rest on a frictionless surface. According to the work-energy theorem, what is the object's final speed?
- About 6.3 meters per second
- About 20 meters per second
- About 40 meters per second
- About 4.5 meters per second
Correct answer: About 6.3 meters per second
The final speed is about 6.3 meters per second. The work-energy theorem states that the net work equals the change in kinetic energy, so 60 joules equals one-half times 3.0 kilograms times v squared; solving gives v squared equals 40, and v equals $\sqrt{40}$ , which is about 6.3 meters per second. The work-energy theorem links the net work directly to the kinetic-energy change, regardless of the path taken.
An object of mass 0.50 kilograms moves in a horizontal circle of radius 2.0 meters at a constant speed of 4.0 meters per second. What is the magnitude of the centripetal force on it?
- 2.0 newtons
- 1.0 newton
- 8.0 newtons
- 4.0 newtons
Correct answer: 4.0 newtons
The centripetal force is 4.0 newtons. The centripetal force formula is F equals m v squared divided by r, so F equals 0.50 times 4.0 squared divided by 2.0 equals 0.50 times 16 divided by 2.0 equals 4.0 newtons. This force always points toward the center of the circle and provides the inward acceleration needed for circular motion.
A 2.0 kilogram cart moving at 3.0 meters per second collides head-on with a stationary 4.0 kilogram cart and they stick together. What is their common velocity immediately after this perfectly inelastic collision?
- 2.0 meters per second
- 1.5 meters per second
- 1.0 meter per second
- 3.0 meters per second
Correct answer: 1.0 meter per second
The common velocity is 1.0 meter per second, which is also the velocity of the center of mass. Momentum is conserved: 2.0 times 3.0 plus 4.0 times 0 equals 6.0 kilogram meters per second, divided by the total mass 6.0 kilograms gives 1.0 meter per second. In a perfectly inelastic collision the objects move together at the center-of-mass velocity, and kinetic energy is not conserved.
Two identical balls undergo a one-dimensional collision: one moving at speed v strikes the other at rest. If the collision is perfectly elastic, what are their velocities afterward?
- Both move together at v/2
- The incoming ball continues at v and the struck ball stays at rest
- Both rebound, each at v/2 in opposite directions
- The incoming ball stops and the struck ball moves off at v
Correct answer: The incoming ball stops and the struck ball moves off at v
In a perfectly elastic collision of equal masses with one initially at rest, the moving ball stops and the struck ball moves off at the original speed v, so the velocities are exchanged. This follows from conserving both momentum and kinetic energy for equal masses. By contrast, in a perfectly inelastic collision the two would move together at v/2 and kinetic energy would be lost.
For a particle moving in a conservative potential, the Euler-Lagrange equation that yields the equations of motion is which of the following, where L is the Lagrangian and q is a generalized coordinate?
- The second time derivative of q equals the partial of L with respect to q
- The time derivative of the partial of L with respect to q-dot, minus the partial of L with respect to q, equals zero
- The time derivative of the partial of L with respect to q, equals the partial of L with respect to q-dot
- The partial of L with respect to q, plus the partial of L with respect to q-dot, equals zero
Correct answer: The time derivative of the partial of L with respect to q-dot, minus the partial of L with respect to q, equals zero
The Euler-Lagrange equation sets the time derivative of the partial of L with respect to q-dot minus the partial of L with respect to q equal to zero. This equation is obtained by requiring the action integral to be stationary, and it generates Newton's equations of motion from the Lagrangian L equals kinetic energy minus potential energy. The quantity inside the time derivative is the generalized momentum conjugate to q.
A single particle of mass m moves in one dimension under a potential V(x) that depends only on position. What is the correct Lagrangian for this system, using x-dot for the velocity?
- V(x) minus one-half m x-dot squared
- One-half m x-dot squared minus V(x)
- One-half m x-dot squared plus V(x)
- M x-dot squared minus V(x)
Correct answer: One-half m x-dot squared minus V(x)
The Lagrangian is one-half m x-dot squared minus V(x), that is, kinetic energy minus potential energy. To find the Lagrangian of a system you write T minus V in terms of generalized coordinates and their time derivatives. The sum T plus V is instead the total mechanical energy, which corresponds to the Hamiltonian for this case, not the Lagrangian.
In Hamiltonian mechanics, the Hamiltonian H for a particle in a time-independent potential most directly represents which physical quantity?
- The difference between kinetic and potential energy
- The total energy, kinetic plus potential, expressed in terms of coordinates and momenta
- The generalized force on the particle
- The action integral over one period of motion
Correct answer: The total energy, kinetic plus potential, expressed in terms of coordinates and momenta
For a particle in a time-independent potential the Hamiltonian represents the total energy, kinetic plus potential, written as a function of the generalized coordinates and conjugate momenta. Hamiltonian mechanics replaces velocities with momenta and uses Hamilton's equations, the partial of H with respect to p gives q-dot and minus the partial of H with respect to q gives p-dot. The difference of kinetic and potential energy is instead the Lagrangian.
A figure skater pulls in her arms while spinning, reducing her moment of inertia to half its initial value. With no external torque, what happens to her angular speed?
- It halves
- It doubles
- It quadruples
- It stays the same
Correct answer: It doubles
Her angular speed doubles. By conservation of angular momentum, the product of moment of inertia and angular velocity stays constant when there is no external torque, so halving the moment of inertia doubles the angular velocity. Her rotational kinetic energy actually increases, supplied by the work she does pulling her arms inward.
Escape velocity from a spherical body of mass M and radius R is derived by setting the kinetic energy equal to the magnitude of the gravitational potential energy at the surface. Which expression results?
- $\sqrt{\frac{GM}{R}}$
- $\sqrt{\frac{2GM}{R}}$
- 2 times G M divided by R
- $\sqrt{\frac{GM}{R^2}}$
Correct answer: $\sqrt{\frac{2GM}{R}}$
The escape velocity is $\sqrt{\frac{2GM}{R}}$ . Setting one-half m v squared equal to G M m divided by R and solving for v cancels the object's mass m and gives v equals $\sqrt{\frac{2GM}{R}}$ . This is why escape velocity is independent of the escaping object's mass and equals $\sqrt{2}$ times the circular orbital speed at the surface.
Using G equal to 6.67 times 10 to the minus 11, Earth's mass 5.97 times 10 to the 24 kilograms, and radius 6.37 times 10 to the 6 meters, what is the approximate escape velocity from Earth's surface?
- About 29.8 kilometers per second
- About 11.2 kilometers per second
- About 16.7 kilometers per second
- About 7.9 kilometers per second
Correct answer: About 11.2 kilometers per second
The escape velocity is about 11.2 kilometers per second. Using v equals $\sqrt{\frac{2GM}{R}}$ , the numerator 2 times 6.67e-11 times 5.97e24 is about 7.97e14, divided by 6.37e6 gives about 1.25e8, and its square root is about 1.12e4 meters per second. The value of about 7.9 kilometers per second is instead the low-Earth-orbit circular speed, smaller by a factor of $\sqrt{2}$ .
Kepler's third law for objects orbiting the same central body states that the square of the orbital period is proportional to the cube of the semi-major axis. If a planet's orbital radius is increased by a factor of 4, by what factor does its period change?
- By a factor of 16
- By a factor of 2
- By a factor of 8
- By a factor of 4
Correct answer: By a factor of 8
The period increases by a factor of 8. Kepler's third law gives T squared proportional to a cubed, so T is proportional to a to the three-halves power; raising a by a factor of 4 multiplies T by 4 to the three-halves power, which equals 8. This is why outer planets have disproportionately longer years than a simple linear scaling would suggest.
A uniform rod of length 1.0 meter swings as a physical pendulum about a horizontal axis through one end, where its moment of inertia is (1/3) M L squared. Taking g equal to 9.8 meters per second squared, what is the period of small oscillations?
- About 1.6 seconds
- About 3.1 seconds
- About 1.3 seconds
- About 2.0 seconds
Correct answer: About 1.6 seconds
The period is about 1.6 seconds. For a physical pendulum the period is 2 pi times $\sqrt{\frac{I}{mgd}}$ ; with I equal to (1/3) M L squared and the center of mass a distance d equal to L/2 from the pivot, this reduces to 2 pi times $\sqrt{\frac{2L}{3g}}$ . Plugging in L equal to 1.0 meter gives 2 pi times $\sqrt{0.068}$ , which is about 1.64 seconds.
A rocket in deep space exhausts gas at an effective exhaust speed of 3000 meters per second. If its initial mass is 3 times its final mass, what is the change in the rocket's speed predicted by the Tsiolkovsky rocket equation?
- About 3000 meters per second
- About 9000 meters per second
- About 1100 meters per second
- About 3300 meters per second
Correct answer: About 3300 meters per second
The change in speed is about 3300 meters per second. The rocket equation, derived from momentum conservation as exhaust is expelled, gives delta v equals the exhaust speed times the natural log of the initial-to-final mass ratio, so delta v equals 3000 times the natural log of 3, which is 3000 times about 1.099, or about 3296 meters per second. The logarithmic dependence on mass ratio is the key result of the derivation.
Two equal masses m are connected in a line by three identical springs of constant k, with the outer springs fixed to walls. What are the angular frequencies of the two normal modes of this coupled oscillator system?
- $\sqrt{\frac{2k}{m}}$ , and $\sqrt{\frac{4k}{m}}$
- $\sqrt{\frac{k}{2m}}$ , and $\sqrt{\frac{3k}{2m}}$
- $\sqrt{\frac{k}{m}}$ , and $\sqrt{\frac{2k}{m}}$
- $\sqrt{\frac{k}{m}}$ , and $\sqrt{\frac{3k}{m}}$
Correct answer: $\sqrt{\frac{k}{m}}$ , and $\sqrt{\frac{3k}{m}}$
The two normal-mode frequencies are $\sqrt{\frac{k}{m}}$ and $\sqrt{\frac{3k}{m}}$ . In the symmetric mode both masses move together, so the central spring is unstretched and each mass feels only one spring, giving omega equal to $\sqrt{\frac{k}{m}}$ . In the antisymmetric mode the masses move oppositely and the central spring is compressed and stretched at double the rate, raising the effective stiffness so omega equals $\sqrt{\frac{3k}{m}}$ .
Faraday's law of induction states that the electromotive force (emf) induced in a closed loop is equal to which of the following?
- The magnetic flux divided by the area enclosed by the loop
- The product of the magnetic flux and the loop's resistance
- The time integral of the magnetic flux through the loop
- The negative rate of change of magnetic flux through the loop
Correct answer: The negative rate of change of magnetic flux through the loop
The induced emf equals the negative rate of change of magnetic flux through the loop. Faraday's law is written emf = -d(Phi_B)/dt, where Phi_B is the magnetic flux. The minus sign encodes Lenz's law: the induced current opposes the change that produced it. A steady (unchanging) flux produces no emf, so it is the time rate of change of flux, not the flux itself, that drives induction.
A long straight wire carries a steady current of 10 amperes. Using the Biot-Savart law, what is the magnitude of the magnetic field at a perpendicular distance of 0.05 meters from the wire? (Use mu_0 = 4 pi x 10^-7 T m/A.)
- 2 x 10^-4 T
- 4 x 10^-5 T
- 1 x 10^-5 T
- 8 x 10^-5 T
Correct answer: 4 x 10^-5 T
The field is 4 x 10^-5 T. Integrating the Biot-Savart law over an infinite straight wire gives B = mu_0 I / (2 pi r). Substituting B = (4 pi x 10^-7)(10) / (2 pi x 0.05) = (2 x 10^-7 x 10) / 0.05 = 2 x 10^-6 / 0.05 = 4 x 10^-5 T. The field circles the wire and falls off as 1/r.
A long air-core solenoid has 1000 turns per meter and carries a current of 2 amperes. What is the approximate magnitude of the magnetic field deep inside the solenoid? (Use mu_0 = 4 pi x 10^-7 T m/A.)
- 5.0 x 10^-3 T
- 8.0 x 10^-4 T
- 2.5 x 10^-3 T
- 1.3 x 10^-3 T
Correct answer: 2.5 x 10^-3 T
The interior field is about 2.5 x 10^-3 T. For an ideal solenoid the field is uniform inside and given by B = mu_0 n I, where n is turns per unit length. Substituting B = (4 pi x 10^-7)(1000)(2) = 8 pi x 10^-4 = 2.51 x 10^-3 T. The field depends only on n and I, not on the solenoid radius.
A resistor of 2000 ohms is in series with a 5 microfarad capacitor and a battery. What is the time constant of this RC circuit?
- 1 millisecond
- 10 milliseconds
- 40 milliseconds
- 100 milliseconds
Correct answer: 10 milliseconds
The time constant is 10 milliseconds. For an RC circuit the time constant is tau = R C, the time for the capacitor voltage to reach about 63 percent of its final value while charging (or decay to 37 percent while discharging). Here tau = (2000)(5 x 10^-6) = 0.01 s = 10 ms. The product of ohms and farads yields seconds by dimensional analysis.
A 4 microfarad capacitor is charged to a potential difference of 200 volts. How much energy is stored in the capacitor?
- 0.04 J
- 0.4 J
- 0.16 J
- 0.08 J
Correct answer: 0.08 J
The stored energy is 0.08 J. The energy in a capacitor is $U = \frac{1}{2} C V^2$ . Substituting $U = \frac{1}{2}(4 \times 10^{-6})(200)^2 = \frac{1}{2}(4 \times 10^{-6})(40000) = 0.08$ J. Equivalent forms $U = \frac{Q^2}{2C}$ and $U = \frac{1}{2} Q V$ give the same result; forgetting the factor of one-half wrongly doubles the answer to 0.16 J.
An inductor of 0.5 henry is placed in series with a 100 ohm resistor and a battery. What is the time constant for the growth of current in this LR circuit?
- 5 milliseconds
- 0.5 milliseconds
- 20 milliseconds
- 50 milliseconds
Correct answer: 5 milliseconds
The time constant is 5 milliseconds. For an LR circuit the time constant is tau = L / R, the time for the current to reach about 63 percent of its final steady value. Substituting tau = 0.5 / 100 = 0.005 s = 5 ms. Note this is L divided by R, unlike the RC case where the time constant is the product R times C.
In a vacuum, the speed of an electromagnetic wave is determined by which combination of the permittivity (epsilon_0) and permeability (mu_0) of free space?
- $\frac{1}{\sqrt{\epsilon_0 \mu_0}}$
- $\sqrt{\epsilon_0 \mu_0}$
- One divided by the product of epsilon_0 and mu_0
- The product of epsilon_0 and mu_0
Correct answer: $\frac{1}{\sqrt{\epsilon_0 \mu_0}}$
The wave speed equals $\frac{1}{\sqrt{\epsilon_0 \mu_0}}$ . Maxwell's equations yield $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ , which evaluates to about $3 \times 10^{8}$ m/s, identifying light as an electromagnetic wave. Substituting the SI values of mu_0 and epsilon_0 reproduces the measured speed of light, a landmark confirmation of electromagnetic theory.
A parallel-plate capacitor has plates of area 0.02 square meters separated by 1.0 millimeter of vacuum. What is its capacitance? (Use epsilon_0 = 8.85 x 10^-12 F/m.)
- 354 pF
- 44 pF
- 89 pF
- 177 pF
Correct answer: 177 pF
The capacitance is about 177 pF. For a parallel-plate capacitor with vacuum between the plates, C = epsilon_0 A / d. Substituting C = (8.85 x 10^-12)(0.02) / (0.001) = (8.85 x 10^-12)(20) = 1.77 x 10^-10 F = 177 pF. Capacitance grows with plate area and shrinks as the gap widens.
A bar magnet is pushed north-pole-first toward a stationary conducting loop. According to Lenz's law, what is the effect of the current induced in the loop?
- It accelerates the magnet toward the loop
- It creates a magnetic field that opposes the approaching magnet, repelling it
- It has no effect on the motion of the magnet
- It creates a magnetic field that attracts the approaching magnet
Correct answer: It creates a magnetic field that opposes the approaching magnet, repelling it
The induced current opposes the approaching magnet and repels it. Lenz's law states that the induced current flows in whatever direction is needed to oppose the change in flux. As the north pole approaches, flux through the loop increases, so the near face of the loop becomes a north pole that pushes back. This opposition is a direct consequence of energy conservation, since the work done against the repulsion supplies the electrical energy.
A single point charge +q sits a distance d in front of a large grounded conducting plane. Using the method of image charges, the field in the region in front of the plane is identical to that produced by the real charge plus what additional fictitious charge?
- A charge -q located a distance d behind the plane
- A charge +2q located at the plane's surface
- A charge -q located a distance 2d behind the plane
- A charge +q located a distance d behind the plane
Correct answer: A charge -q located a distance d behind the plane
The configuration is reproduced by adding an image charge -q a distance d behind the plane. The method of image charges replaces the grounded conductor with a single mirror charge of opposite sign at the mirror-image position, which automatically makes the plane an equipotential at zero potential. The real charge is then attracted to the plane with a force equal to that between +q and -q separated by 2d.
Gauss's law for the electric field relates the net electric flux through a closed surface to which quantity?
- The total electric charge enclosed by the surface divided by epsilon_0
- The rate of change of magnetic flux through the surface
- The total current passing through the surface
- The total charge on the surface multiplied by epsilon_0
Correct answer: The total electric charge enclosed by the surface divided by epsilon_0
The net electric flux equals the enclosed charge divided by epsilon_0. Gauss's law states that the surface integral of E over a closed surface equals Q_enclosed / epsilon_0. Charges outside the surface contribute zero net flux because their field lines enter and exit. This law is one of Maxwell's four equations and is most powerful for problems with high symmetry, such as spheres, cylinders, and planes.
Which statement correctly characterizes the four Maxwell's equations of classical electromagnetism?
- They require the existence of isolated magnetic monopoles
- They include Gauss's law for electricity, Gauss's law for magnetism, Faraday's law, and the Ampere-Maxwell law
- They state that both electric and magnetic field lines always begin and end on charges
- They apply only to static fields and cannot describe wave propagation
Correct answer: They include Gauss's law for electricity, Gauss's law for magnetism, Faraday's law, and the Ampere-Maxwell law
Maxwell's equations consist of Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of induction, and the Ampere-Maxwell law. Gauss's law for magnetism states there are no magnetic monopoles, so magnetic field lines form closed loops. Together with the displacement-current term Maxwell added to Ampere's law, these equations predict self-propagating electromagnetic waves traveling at the speed of light.
For an electromagnetic wave, the Poynting vector S describes the flow of electromagnetic energy. Which expression gives the Poynting vector?
- The cross product of E and B divided by mu_0
- The cross product of E and B times epsilon_0
- The dot product of E and B times mu_0
- E squared divided by mu_0
Correct answer: The cross product of E and B divided by mu_0
The Poynting vector is S = (1/mu_0)(E cross B). It points in the direction of energy flow (the propagation direction for a plane wave) and its magnitude gives the instantaneous power per unit area. For a sinusoidal plane wave the time-averaged magnitude, the intensity, is $\frac{1}{2} \epsilon_0 c E_0^2$ , where $E_0$ is the electric field amplitude.
At a point far from an electric dipole, how does the electric potential vary with the distance r from the dipole?
- It falls off as 1 over r cubed
- It is independent of r
- It falls off as 1 over r
- It falls off as 1 over r squared
Correct answer: It falls off as 1 over r squared
The dipole potential falls off as 1 over r squared. The potential of a dipole is $V = \frac{1}{4\pi\epsilon_0} \frac{p\cos\theta}{r^2}$ , where p is the dipole moment and theta is measured from the dipole axis. This faster falloff than a single point charge (which goes as 1/r) arises because the equal and opposite charges nearly cancel at large distances. The dipole field itself then falls off as 1 over r cubed.
An infinitely long straight line of charge has a uniform linear charge density $\lambda = 2 \times 10^{-9}$ C/m. What is the magnitude of the electric field at a perpendicular distance of 0.1 meters from the line? (Use $\frac{1}{4\pi\epsilon_0} = 9 \times 10^{9}$ N m $^2$ /C $^2$ .)
- 720 V/m
- 360 V/m
- 180 V/m
- 90 V/m
Correct answer: 360 V/m
The field is about 360 V/m. For an infinite line charge, Gauss's law gives $E = \frac{\lambda}{2\pi\epsilon_0 r}$ , which equals $\frac{2k\lambda}{r}$ with $k = \frac{1}{4\pi\epsilon_0}$ . Substituting $E = \frac{2(9 \times 10^{9})(2 \times 10^{-9})}{0.1} = \frac{36}{0.1} = 360$ V/m. The field points radially away from the line and falls off as 1/r, slower than a point charge.
Ampere's law (in its static form) is most directly useful for computing the magnetic field in which of the following situations?
- The field of a single moving point charge
- The field produced by a time-varying electric flux alone
- The field around configurations with high symmetry, such as a long straight wire or a solenoid
- The field of an arbitrary irregular current distribution
Correct answer: The field around configurations with high symmetry, such as a long straight wire or a solenoid
Ampere's law is most useful for highly symmetric current configurations such as a long straight wire, a solenoid, or a toroid. The law states that the line integral of B around a closed loop equals mu_0 times the enclosed current. As with Gauss's law, the field can be pulled out of the integral only when symmetry makes its magnitude constant along the chosen Amperian loop; otherwise the Biot-Savart law is needed instead.
At the interface between two dielectrics with no free surface charge, which component of the electric field is continuous across the boundary?
- The tangential component of E
- The normal component of E
- The normal component of the displacement field D is discontinuous
- The magnitude of E is always continuous
Correct answer: The tangential component of E
The tangential component of E is continuous across a dielectric boundary. This follows from the curl-free condition on the static electric field (Faraday's law with no changing magnetic flux). The normal component of E is generally discontinuous because bound surface charge appears at the interface; it is instead the normal component of the displacement field D that is continuous when there is no free surface charge.
A particle of charge q moves with velocity v through a region containing electric field E and magnetic field B. What is the total electromagnetic (Lorentz) force on the particle?
- Q times E only
- Q times (v cross B) only
- Q times E plus q times (B cross v) times the speed of light
- Q times E plus q times (v cross B)
Correct answer: Q times E plus q times (v cross B)
The Lorentz force is F = qE + q(v cross B). The electric part acts along E regardless of motion, while the magnetic part is perpendicular to both v and B and therefore does no work on the particle, changing only its direction. Because the magnetic force is always perpendicular to the velocity, a charged particle in a uniform magnetic field moves in a circle or helix at constant speed.
A thin converging lens has a focal length of 20 cm. An object is placed 30 cm in front of the lens on the optical axis. Where is the image formed, and what is its lateral magnification?
- 60 cm behind the lens, magnification -2.0
- 12 cm in front of the lens, magnification +0.4
- 60 cm in front of the lens, magnification +2.0
- 15 cm behind the lens, magnification -0.5
Correct answer: 60 cm behind the lens, magnification -2.0
The image forms 60 cm behind the lens with magnification -2.0, meaning it is real, inverted, and twice the object size. Using the thin lens equation 1/v - 1/u = 1/f with the convention u = -30 cm and f = +20 cm: 1/v = 1/f + 1/u = 1/20 - 1/30 = 1/60, so v = +60 cm (positive means a real image on the far side). The lateral magnification is m = -v/u = -(60)/(30) = -2.0, where the negative sign indicates an inverted image. A virtual upright image only occurs when the object lies inside the focal length.
In a Young's double-slit experiment, two slits separated by 0.10 mm are illuminated with light of wavelength 600 nm, and the interference pattern is observed on a screen 2.0 m away. What is the spacing between adjacent bright fringes?
- 3.0 cm
- 1.2 cm
- 0.60 cm
- 0.30 cm
Correct answer: 1.2 cm
Adjacent bright fringes are spaced 1.2 cm apart. For double-slit interference the fringe spacing on the screen is delta_y = lambda*L/d, where L is the slit-to-screen distance and d is the slit separation. Substituting delta_y = (600e-9 m)(2.0 m)/(1.0e-4 m) = 1.2e-2 m = 1.2 cm. A spacing of 0.60 cm would result from mistakenly using half the wavelength or doubling the slit separation; the bright-fringe condition is d*sin(theta) = m*lambda for integer m.
A fixed quantity of an ideal gas is held in a rigid sealed container. The absolute temperature of the gas is raised from 300 K to 600 K. By what factor does the pressure of the gas change?
- It is multiplied by 2
- It stays the same
- It is multiplied by $\sqrt{2}$
- It is multiplied by 4
Correct answer: It is multiplied by 2
The pressure is multiplied by 2. The ideal gas law PV = nRT, with V, n and R all constant, forces P to be directly proportional to absolute temperature T, so doubling T from 300 K to 600 K doubles the pressure. The factor-of-4 distractor wrongly assumes a square dependence, which would apply to neither pressure nor temperature here.
A Carnot engine operates between a hot reservoir at 500 K and a cold reservoir at 300 K. What is its maximum theoretical efficiency?
- 0.83
- 0.60
- 0.20
- 0.40
Correct answer: 0.40
The efficiency is 0.40, or 40 percent. The Carnot efficiency equals 1 minus the ratio of cold to hot absolute temperatures, which is 1 minus 300/500 = 1 minus 0.60 = 0.40. The 0.60 distractor mistakenly reports the temperature ratio 300/500 itself rather than one minus that ratio.
The surface of a star behaves approximately as a blackbody and emits radiation that peaks at a wavelength of about 580 nm. According to Wien's displacement law, what happens to the peak wavelength if the surface temperature of the star doubles?
- The peak wavelength is multiplied by 4
- The peak wavelength is doubled
- The peak wavelength stays the same
- The peak wavelength is halved
Correct answer: The peak wavelength is halved
The peak wavelength is halved. Wien's displacement law states that the peak emission wavelength is inversely proportional to absolute temperature (lambda_max times T equals a constant, about 2.90 times 10 to the minus 3 meter-kelvin), so doubling the temperature shifts the peak to half the original wavelength, toward the blue. The distractor that doubles the wavelength inverts the correct inverse relationship.
For a sample of an ideal monatomic gas in thermal equilibrium, how does the root-mean-square molecular speed compare with the most probable molecular speed of the Maxwell-Boltzmann distribution?
- The rms speed is larger, by exactly a factor of 2
- The rms speed is smaller, by a factor equal to $\sqrt{\frac{2}{3}}$ (about 0.82)
- The two speeds are exactly equal
- The rms speed is larger, by a factor equal to $\sqrt{\frac{3}{2}}$ (about 1.22)
Correct answer: The rms speed is larger, by a factor equal to $\sqrt{\frac{3}{2}}$ (about 1.22)
The rms speed exceeds the most probable speed by $\sqrt{\frac{3}{2}}$ , about 1.22. In the Maxwell-Boltzmann distribution the rms speed equals $\sqrt{\frac{3kT}{m}}$ while the most probable speed equals $\sqrt{\frac{2kT}{m}}$ , so their ratio is $\sqrt{\frac{3}{2}}$ . The ordering is always most probable speed less than mean speed less than rms speed, which rules out the equal and smaller-rms options.
In a system of identical fermions described by Fermi-Dirac statistics at a nonzero temperature, what is the average occupation number of a single-particle energy level whose energy exactly equals the chemical potential (Fermi energy)?
- Exactly 0
- Exactly 1
- Exactly 1/2
- It depends on the temperature, ranging from 0 to 1
Correct answer: Exactly 1/2
The average occupation is exactly 1/2. The Fermi-Dirac distribution gives an occupation of 1 divided by the quantity (exponential of (E minus mu) over kT, plus 1); when E equals the chemical potential mu, the exponent is zero, the exponential is 1, and the occupation becomes 1 over (1 plus 1), which is 1/2 at every temperature. This is why the chemical potential is the half-occupancy energy, and the temperature-dependent distractor misses that this particular level is fixed at 1/2.
An electron is accelerated from rest through a potential difference of 100 volts. Using the de Broglie relation lambda = h/p with non-relativistic momentum, which expression gives the electron's de Broglie wavelength in terms of its kinetic energy E?
- Lambda = h divided by (m E)
- Lambda = (2 m E) divided by h
- $\lambda = \frac{h}{\sqrt{2mE}}$
- Lambda = h E divided by (2 m)
Correct answer: $\lambda = \frac{h}{\sqrt{2mE}}$
The de Broglie wavelength is $\lambda = \frac{h}{\sqrt{2mE}}$ . The de Broglie relation is $\lambda = \frac{h}{p}$ , and for a non-relativistic particle the kinetic energy $E = \frac{p^2}{2m}$ , so $p = \sqrt{2mE}$ . Substituting gives $\lambda = \frac{h}{\sqrt{2mE}}$ ; plugging in 100 eV for an electron yields about 0.12 nanometers. The expressions with E in the numerator have the wrong dimensions and fail dimensional analysis.
A measurement determines a particle's momentum with an uncertainty of essentially zero. According to the Heisenberg uncertainty principle, what does this imply about the uncertainty in the particle's position?
- Both uncertainties settle at the value h-bar over 2
- The position uncertainty becomes infinitely large
- The position uncertainty also approaches zero
- The position uncertainty equals the momentum value divided by h-bar
Correct answer: The position uncertainty becomes infinitely large
The position uncertainty becomes infinitely large. The Heisenberg uncertainty principle states that the product of the uncertainties in position and momentum satisfies delta-x times delta-p greater than or equal to h-bar over 2. If delta-p approaches zero, then delta-x must approach infinity to keep the product at or above h-bar over 2. This is why a state of perfectly definite momentum, a plane wave, is spread out over all space.
For a particle confined to a one-dimensional infinite square well of width L, the energy levels are E sub n proportional to n squared. By what factor does the energy of the third excited state (n = 4) exceed the ground-state energy (n = 1)?
- 8
- 4
- 12
- 16
Correct answer: 16
The factor is 16. The infinite square well energy levels are E sub n = n squared times h squared over (8 m L squared), so each level scales as n squared. The third excited state has n = 4 (the states are n = 1, 2, 3, 4 for ground, first, second, and third excited), giving 4 squared = 16 times the ground-state energy. A common error is choosing n = 3, which would give a factor of 9, or counting the third excited state as n = 3.
The quantized energy levels of a one-dimensional quantum harmonic oscillator are given by E sub n = (n + 1/2) times h-bar times omega, for n = 0, 1, 2, and so on. What is the energy spacing between any two adjacent levels?
- (2n + 1) times h-bar times omega
- It increases with increasing n
- h-bar times omega
- One-half times h-bar times omega
Correct answer: h-bar times omega
The spacing is h-bar times omega. Because E sub n = (n + 1/2) times h-bar times omega, the difference between level n+1 and level n is exactly h-bar times omega, independent of n, so the levels are evenly spaced. This uniform spacing is a hallmark of the harmonic oscillator and contrasts with the infinite square well, whose spacing grows with n. The ground-state energy is one-half h-bar omega (the zero-point energy), but that is not the spacing between adjacent levels.
A particle is in the ground state (n = 1) of a one-dimensional infinite square well of width L, with wavefunction proportional to the sine of (pi x over L). At which location is the particle most likely to be found?
- At the center of the well, x = L over 2
- At x = L over 4 and x = 3L over 4
- With equal probability everywhere in the well
- At either wall of the well
Correct answer: At the center of the well, x = L over 2
The particle is most likely found at the center, x = L over 2. The ground-state wavefunction is proportional to sine of (pi x over L), and the probability density is its square, which peaks where the sine equals one, namely at x = L over 2. The probability density vanishes at the walls because the wavefunction must be zero there. Uniform probability everywhere describes a classical particle, not the quantum ground state.
A particle of energy E encounters a rectangular potential barrier of height V (with V greater than E) and width L. Within the standard approximation, the tunneling transmission probability behaves as T proportional to $e^{-2\kappa L}$ . What happens to the transmission probability if the barrier width L is doubled?
- It is squared
- It is reduced by a factor of two
- It increases linearly with width
- It is unchanged because tunneling depends only on energy
Correct answer: It is squared
The transmission probability is squared. Since T is proportional to e raised to minus 2 kappa L, doubling L replaces the exponent with minus 2 kappa times (2L), which is the original exponential factor squared. Because that factor is less than one, squaring it makes the transmission much smaller, showing tunneling falls off exponentially, not linearly, with barrier width. Tunneling clearly does depend on barrier width, so the unchanged option is wrong.
For a quantum system in a normalized state described by wavefunction psi(x), the expectation value of an observable represented by the operator A-hat is computed as which of the following?
- The square of the integral of psi over all x
- The integral of A-hat times psi, over all x
- The value of A-hat evaluated at the most probable position
- The integral of psi-star times A-hat acting on psi, over all x
Correct answer: The integral of psi-star times A-hat acting on psi, over all x
The expectation value is the integral over all space of psi-star times A-hat acting on psi. This sandwich form, with the complex conjugate psi-star on the left and the operator acting on psi on the right, gives the probability-weighted average of the observable for the state. For a normalized state the result is the mean value one would obtain by averaging many identical measurements. Simply evaluating the operator at the most probable position ignores the full probability distribution and is incorrect.
The time-independent Schrodinger equation for a particle of mass m in a potential V(x) can be written as H-hat acting on psi equals E psi. What does this equation physically represent?
- A requirement that the wavefunction be a real-valued function
- An eigenvalue equation whose solutions are stationary states with definite energy E
- A statement that energy is not conserved in quantum systems
- A formula giving the particle's exact position as a function of time
Correct answer: An eigenvalue equation whose solutions are stationary states with definite energy E
It is an eigenvalue equation whose solutions are stationary states of definite energy E. The Hamiltonian operator H-hat represents total energy, and the equation H-hat psi = E psi picks out the special wavefunctions (eigenfunctions) for which acting with the energy operator simply returns the same function times a number E (the eigenvalue, the allowed energy). These stationary states have a probability density that does not change in time. Wavefunctions are generally complex, so requiring them to be real is incorrect.
Which pairs of observables in quantum mechanics can in principle be measured simultaneously with arbitrary precision?
- Position and momentum along the same axis
- Observables whose operators commute
- Energy and time
- Any two observables, since precision is limited only by the apparatus
Correct answer: Observables whose operators commute
Observables whose operators commute can be measured simultaneously with arbitrary precision. When two operators commute, their commutator is zero, and they share a common set of eigenstates, so a system can have definite values of both at once. Position and momentum along the same axis do not commute (their commutator equals i times h-bar), which is exactly why the uncertainty principle limits their joint precision. The limitation is fundamental, not merely a matter of apparatus quality.
An electron and a proton are accelerated so that each has the same kinetic energy in the non-relativistic regime. How does the de Broglie wavelength of the electron compare to that of the proton?
- The electron's wavelength is shorter because it moves faster
- The electron's wavelength is longer because it has the smaller mass
- Their wavelengths are equal because their energies are equal
- The electron's wavelength is longer because it carries negative charge
Correct answer: The electron's wavelength is longer because it has the smaller mass
The electron's wavelength is longer because it has the smaller mass. For equal kinetic energy E, the de Broglie wavelength is $\lambda = \frac{h}{\sqrt{2mE}}$ , so a smaller mass gives a longer wavelength. The proton is roughly 1836 times heavier than the electron, making its momentum larger at the same energy and its wavelength correspondingly shorter. Charge sign has no effect on the de Broglie wavelength, which depends only on momentum.
The Rydberg formula gives the inverse wavelength of a hydrogen spectral line as 1/lambda = R times (1/n_f squared minus 1/n_i squared), where R is approximately 1.097 times 10 to the 7th per meter. For the Balmer-series line produced by the transition from n = 3 down to n = 2, what is the approximate wavelength of the emitted photon?
- 410 nm
- 656 nm
- 122 nm
- 1875 nm
Correct answer: 656 nm
656 nm is correct. Substituting n_f = 2 and n_i = 3 gives 1/lambda = R times (1/4 minus 1/9) = 1.097e7 times (5/36) = 1.524e6 per meter, so lambda = 1/(1.524e6) = 6.56e-7 m, about 656 nm (the red H-alpha line). The 122 nm value is the Lyman-alpha line (n = 2 to n = 1), and 1875 nm is the Paschen-series first line (n = 4 to n = 3), so those use the wrong final level.
In Compton scattering, a photon scatters off a free electron and its wavelength increases by an amount Delta-lambda = (h / m_e c) times (1 minus cosine theta), where theta is the scattering angle. For a given incident photon, at what scattering angle is the wavelength shift the largest, and what is the maximum shift in terms of the electron Compton wavelength lambda_C = h / m_e c?
- At 0 degrees, with maximum shift equal to lambda_C
- At 180 degrees, with maximum shift equal to 2 times lambda_C
- At 180 degrees, with maximum shift equal to lambda_C
- At 90 degrees, with maximum shift equal to lambda_C
Correct answer: At 180 degrees, with maximum shift equal to 2 times lambda_C
At 180 degrees with a maximum shift of 2 times lambda_C is correct. The factor (1 minus cosine theta) is largest when theta = 180 degrees (backscatter), where cosine theta = minus 1, giving (1 minus (minus 1)) = 2, so Delta-lambda = 2 lambda_C. At theta = 0 degrees the photon passes straight through and there is zero shift, and at 90 degrees the shift equals exactly one lambda_C, not the maximum.
The Compton wavelength of the electron, defined as h divided by the product of the electron mass and the speed of light, represents the wavelength of a photon whose energy equals the electron rest energy. What is its approximate numerical value?
- 0.529 pm
- 2.43 pm
- 656 nm
- 91.2 nm
Correct answer: 2.43 pm
2.43 pm (2.43 times 10 to the minus 12 meter, or 0.00243 nm) is correct. Using h = 6.63e-34 joule-second, m_e = 9.11e-31 kilogram, and c = 3.00e8 meter per second, lambda_C = h/(m_e c) = 2.43e-12 m. The 0.529 pm value is a plausible-looking decoy, and 91.2 nm is the hydrogen Lyman series limit, neither of which is the electron Compton wavelength.
The fine-structure constant alpha characterizes the strength of the electromagnetic interaction and, in Gaussian units, is written as alpha = e squared divided by (h-bar times c). Which statement about this constant is correct?
- It is dimensionless and approximately equal to 137
- It is dimensionless and approximately equal to 1/137
- It has units of energy and approximately equals 13.6 eV
- It has units of inverse length and approximately equals 1.097 times 10 to the 7th per meter
Correct answer: It is dimensionless and approximately equal to 1/137
Dimensionless and approximately equal to 1/137 is correct. The combination e squared over (h-bar c) has the charge, action, and speed units cancel, leaving a pure number near 0.00729, which equals 1/137.036. It is not 137 itself (that is its reciprocal), and it carries no units, so the energy and inverse-length choices are wrong. Its small size is why electromagnetism is a relatively weak coupling and why atomic fine-structure splittings are small.
In the photoelectric effect, monochromatic light of frequency f strikes a metal whose work function is phi. According to the Einstein photoelectric equation, what is the maximum kinetic energy of the ejected photoelectrons?
- H times f minus phi
- Phi minus h times f
- H times f plus phi
- H times f divided by phi
Correct answer: H times f minus phi
h times f minus phi is correct. Each photon delivers energy h f; the work function phi is the minimum energy needed to liberate an electron from the surface, so by energy conservation the maximum kinetic energy is the leftover, K_max = h f minus phi. If h f is less than phi no electrons are emitted regardless of light intensity. Adding the two quantities or dividing them does not conserve energy and is dimensionally or physically wrong.
The bound-state energy levels of the hydrogen atom are given by E_n = minus 13.6 eV divided by n squared, where n is the principal quantum number. What is the energy required to ionize a hydrogen atom that is already in its first excited state (n = 2)?
- 27.2 eV
- 13.6 eV
- 3.4 eV
- 10.2 eV
Correct answer: 3.4 eV
3.4 eV is correct. The n = 2 level sits at E_2 = minus 13.6/4 = minus 3.4 eV. Ionization means raising the electron to E = 0 (n approaching infinity), so the energy needed is 0 minus (minus 3.4 eV) = 3.4 eV. The 13.6 eV value is the ionization energy from the ground state (n = 1), and 10.2 eV is the n = 1 to n = 2 excitation energy, not the ionization energy from n = 2.
In the Bohr model of a hydrogen-like atom, the orbital radius scales as r_n = n squared times a_0 divided by Z, where a_0 = 0.529 angstrom is the Bohr radius, n is the principal quantum number, and Z is the nuclear charge. What is the radius of the n = 3 orbit in a neutral hydrogen atom (Z = 1)?
- 4.76 angstrom
- 1.59 angstrom
- 0.529 angstrom
- 2.12 angstrom
Correct answer: 4.76 angstrom
4.76 angstrom is correct. For hydrogen Z = 1, so r_3 = 3 squared times a_0 = 9 times 0.529 = 4.76 angstrom. The radius grows with the square of n, not linearly, so the 1.59 angstrom result (which is 3 times a_0) and the 2.12 angstrom result (4 times a_0) both misapply the scaling. The 0.529 angstrom value is the ground-state (n = 1) radius.
When atoms are placed in an external magnetic field, individual spectral lines split into several closely spaced components, an effect named after its discoverer. As the strength of the applied magnetic field is increased, how does the spacing between adjacent split components behave?
- It oscillates periodically with field strength
- It increases in proportion to the magnetic field strength
- It decreases as the field strength increases
- It stays constant, independent of field strength
Correct answer: It increases in proportion to the magnetic field strength
It increases in proportion to the magnetic field strength is correct. This is the Zeeman effect: the magnetic field shifts each magnetic sublevel by an energy equal to the magnetic moment projection times the field, so the energy splitting between adjacent sublevels grows linearly with B. With zero applied field the components merge back into a single unsplit line, which rules out a constant or field-independent spacing.
Electric-dipole selection rules govern which radiative transitions between atomic states are allowed. For a single electron transition involving the orbital angular momentum quantum number l, which change in l corresponds to an allowed electric-dipole transition?
- Any value of Delta-l is allowed
- Delta-l equals 0
- Delta-l equals plus or minus 2
- Delta-l equals plus or minus 1
Correct answer: Delta-l equals plus or minus 1
Delta-l equals plus or minus 1 is correct. An emitted or absorbed dipole photon carries one unit of orbital angular momentum, so conservation requires the electron's l to change by exactly one unit, for example a 3p-to-2s or 2p-to-1s transition. A transition with Delta-l = 0 (such as s-to-s) or Delta-l = plus or minus 2 (such as d-to-s) is forbidden as an electric-dipole process and at most proceeds far more weakly through higher-order multipole mechanisms.
The Lyman series of hydrogen consists of all transitions ending on the n = 1 level. Using the Rydberg constant R = 1.097 times 10 to the 7th per meter, the series limit (the shortest wavelength in the series, corresponding to a transition from n = infinity down to n = 1) is approximately which value?
- 122 nm
- 656 nm
- 365 nm
- 91 nm
Correct answer: 91 nm
The Lyman series limit is about 91 nm. At the series limit n_i goes to infinity, so 1/lambda = R times (1/1 minus 0) = R, giving lambda = 1/R = 1/(1.097 times 10 to the 7th) which equals roughly 9.1 times 10 to the minus 8th meters, or 91 nm, in the far ultraviolet. The 122 nm value is the longest Lyman line (n = 2 to n = 1), not the limit, and 656 nm is the Balmer alpha line.
An atom emits a spectral line that splits into more than three components when placed in a weak external magnetic field, and the component spacings are not equal multiples of a single value. This pattern, which could not be explained until electron spin was introduced, is best described as which effect?
- The normal Zeeman effect
- The anomalous Zeeman effect
- The photoelectric effect
- The Compton effect
Correct answer: The anomalous Zeeman effect
This is the anomalous Zeeman effect. It occurs for transitions between levels with nonzero total spin, where the upper and lower levels have different Lande g-factors, producing more than three lines with unequal spacing. The normal Zeeman effect, by contrast, gives exactly three equally spaced lines and occurs only for spin-singlet (S = 0) levels; explaining the anomalous pattern historically required the concept of electron spin.
In the 1922 Stern-Gerlach experiment, a beam of neutral silver atoms passed through a strongly inhomogeneous magnetic field and split into exactly two distinct beams on the detector. What property of the atom's unpaired valence electron does this two-fold splitting demonstrate?
- Its electric charge
- Its orbital angular momentum quantum number l
- Its principal quantum number n
- Its spin angular momentum, which has two allowed orientations
Correct answer: Its spin angular momentum, which has two allowed orientations
The two-beam result demonstrates electron spin angular momentum, which has exactly two allowed projections along the field (spin up and spin down). Silver's single unpaired valence electron is in an l = 0 (s) state, so it carries no orbital angular momentum to produce splitting; the deflection into two beams instead reveals the intrinsic spin one-half nature of the electron, with magnetic quantum number m_s equal to plus or minus one-half.
Moseley found that the characteristic K-alpha X-ray emission of an element follows a simple regularity. If the square root $\sqrt{\nu}$ of the K-alpha frequency is plotted against atomic number Z, the data fall on a straight line. This relationship is most useful because it allows one to determine which atomic quantity directly?
- The work function of the metal
- The number of neutrons in the nucleus
- The orbital eccentricity of the electron
- The atomic number Z of an element
Correct answer: The atomic number Z of an element
Moseley's law lets one determine the atomic number Z directly. The square root $\sqrt{\nu}$ of the K-alpha frequency is proportional to (Z minus 1), where the constant 1 accounts for screening of the nuclear charge by the remaining K-shell electron. Because the relation depends on nuclear charge rather than atomic weight, Moseley's law fixed the correct ordering of elements in the periodic table and showed that Z, not atomic mass, is the fundamental ordering quantity.
In the Bohr model, the electron's orbital angular momentum is quantized in integer multiples of h-bar (Planck's constant divided by 2 pi). For an electron in the n = 3 Bohr orbit of hydrogen, what is the magnitude of its orbital angular momentum?
- 3 times h-bar
- One-half times h-bar
- h-bar divided by 3
- 9 times h-bar
Correct answer: 3 times h-bar
The angular momentum is 3 times h-bar. Bohr postulated L = n times h-bar, so for n = 3 the orbital angular momentum equals 3 h-bar. This quantization condition is equivalent to requiring an integer number of de Broglie wavelengths around the circular orbit; note that the radius scales as n squared while the angular momentum scales linearly with n.
An electron in a hydrogen atom is in a state with principal quantum number n = 3. Ignoring fine structure and electron spin, how many distinct (n, l, m_l) orbital states share this same energy level?
- 6
- 1
- 9
- 3
Correct answer: 9
There are 9 degenerate orbital states for n = 3. In hydrogen the energy depends only on n, and for a given n the allowed orbital quantum numbers are l = 0, 1, up to n minus 1, with each l having 2l plus 1 values of m_l. Summing 1 + 3 + 5 for l = 0, 1, 2 gives 9 states (the n squared degeneracy); including the two spin states would double this to 18.
A physicist must explain two distinct experiments: in the first, ultraviolet light ejects electrons from a metal surface only above a threshold frequency; in the second, X-rays scatter off electrons and emerge with a longer wavelength. Which pairing correctly identifies the dominant photon-electron interaction in each case?
- Photoelectric effect for both
- Compton scattering for the first, photoelectric effect for the second
- Pair production for the first, photoelectric effect for the second
- Photoelectric effect for the first, Compton scattering for the second
Correct answer: Photoelectric effect for the first, Compton scattering for the second
The first is the photoelectric effect and the second is Compton scattering. In the photoelectric effect a photon is fully absorbed by a bound electron, ejecting it only when the photon energy h f exceeds the work function. In Compton scattering a higher-energy photon scatters off a nearly free electron, transferring part of its energy and emerging with a longer wavelength shifted by (h / m_e c) times (1 minus cosine theta). The photoelectric effect dominates at lower photon energies while Compton scattering dominates at X-ray energies.
A beam of X-rays with wavelength 0.150 nm strikes a crystal whose atomic planes are separated by 0.300 nm. At approximately what glancing angle (measured from the plane) does the first-order Bragg reflection appear?
- About 14.5 degrees
- About 7.2 degrees
- About 45 degrees
- About 30 degrees
Correct answer: About 14.5 degrees
About 14.5 degrees is correct. Bragg's law states n times lambda equals 2 d sine theta; for first-order reflection n equals 1, so sine theta equals lambda divided by 2d, which is 0.150 nm divided by 0.600 nm, equal to 0.25. The angle whose sine is 0.25 is about 14.5 degrees. The 30-degree value mistakenly drops the factor of 2 in the plane spacing.
On the curve of nuclear binding energy per nucleon versus mass number, the maximum occurs near iron (mass number about 56). What does this peak imply about how energy can be released?
- Both fusion and fission of iron-56 release energy
- Fusion of nuclei lighter than iron releases energy, while fission of nuclei heavier than iron releases energy
- Only fusion releases energy, regardless of the starting nucleus
- Only fission releases energy, regardless of the starting nucleus
Correct answer: Fusion of nuclei lighter than iron releases energy, while fission of nuclei heavier than iron releases energy
Fusion of nuclei lighter than iron releases energy, while fission of nuclei heavier than iron releases energy is correct. Iron-56 has nearly the highest binding energy per nucleon (about 8.8 MeV), so it is among the most tightly bound nuclei. Light nuclei reach higher binding energy per nucleon by fusing, and heavy nuclei reach higher binding energy per nucleon by splitting, so both processes move toward iron and release energy. Iron itself sits at the peak, so neither fusing nor fissioning it releases energy.
A radioactive isotope has a half-life of 6.0 days. A freshly prepared sample contains a certain number of these nuclei. After how many days will only 12.5 percent of the original nuclei remain?
- 24 days
- 12 days
- 18 days
- 48 days
Correct answer: 18 days
18 days is correct. The fraction remaining after n half-lives is (1/2) raised to the n; 12.5 percent equals 1/8, which equals (1/2) cubed, so n equals 3 half-lives. Three half-lives of 6.0 days each is 18 days. A value of 24 days corresponds to four half-lives, which would leave 6.25 percent, not 12.5 percent.
In an X-ray tube, electrons are accelerated through a potential difference of 50 kV before striking a metal target. The continuous bremsstrahlung spectrum produced has a sharp short-wavelength cutoff. What is the approximate minimum wavelength of the emitted X-rays?
- About 0.25 nm
- About 2.5 nm
- About 0.025 nm
- About 0.0050 nm
Correct answer: About 0.025 nm
About 0.025 nm is correct. Bremsstrahlung arises when accelerated electrons decelerate in the target; the most energetic photon carries the electron's full kinetic energy, giving the Duane-Hunt cutoff lambda-min equals hc divided by eV. Using hc equal to 1240 eV-nm and eV equal to 50000 eV gives 1240 divided by 50000, about 0.025 nm. A value of 0.25 nm errs by a factor of ten in the accelerating voltage.
A solid behaves as an electrical insulator at low temperature because of its electronic band structure. Which statement best describes the band arrangement responsible for insulating behavior?
- A partially filled conduction band that allows electrons to move freely
- A conduction band that is completely empty with no valence electrons present
- A completely filled valence band separated from an empty conduction band by a large energy gap
- Overlapping valence and conduction bands with no energy gap
Correct answer: A completely filled valence band separated from an empty conduction band by a large energy gap
A completely filled valence band separated from an empty conduction band by a large energy gap is correct. In an insulator the valence band is full, so electrons have no nearby empty states to accept the small energy from an applied field, and the wide band gap (typically several electron volts) prevents thermal excitation into the conduction band at ordinary temperatures. A partially filled band describes a metal, and overlapping bands also describe metallic or semimetallic conduction.

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By Career EmployerUpdated Jun 7, 2026

GRE Physics Question of the DayNew daily

1 of 160+ free questions — a new one every day

Which principle states that no two fermions can occupy the same quantum state?

Pick an answer to see the explanation

Click Start Test above to launch a full-length GRE Physics practice test weighted like the real ETS exam, or drill a single topic — Classical Mechanics, Electromagnetism, Quantum Mechanics, and beyond. Every question includes a clear explanation so you learn the reasoning, not just the answer.

The GRE Physics Subject Test is administered by ETS (Educational Testing Service) and used by graduate physics programs to compare applicants from different undergraduate backgrounds.

^[1] It is an admissions test, not a pass/fail certification — your total score is reported on a scaled 200–990 range with a percentile rank. These free GRE Physics practice questions mirror the official ETS content distribution.

^[2] To round out your prep, pair these with our free study guide, flashcards.

GRE Physics at a Glance

GRE Physics Subject Test at a glance
Detail	GRE Physics Subject Test
Administered by	ETS (Educational Testing Service)
Questions	About 100 five-option multiple-choice
Time limit	2 hours 50 minutes (170 minutes)
Format	Computer-delivered, multiple choice
Scoring	Scaled 200–990 (10-point increments) + percentile; no pass/fail
Subscores	Percent-correct: Mechanics, E&M, Quantum/Atomic
Purpose	Graduate physics program admissions
Registration fee	$150 (verify current ETS fee)

What Is on the GRE Physics Test?

The GRE Physics Subject Test covers nine content areas: Classical Mechanics (20%), Electromagnetism (18%), Quantum Mechanics (12%), Atomic Physics (10%), Thermodynamics and Statistical Mechanics (10%), Optics and Wave Phenomena (9%), Specialized Topics (9%), Special Relativity (6%), and Laboratory Methods (6%).^[1]

ETS sets these approximate percentages based on a nationwide survey of undergraduate physics curricula, with Classical Mechanics the largest area. Our full practice test is weighted to match ETS’s published content distribution:

GRE Physics weighting by content area (ETS)

Classical Mechanics20% · ≈13 Qs

Electromagnetism18% · ≈12 Qs

Quantum Mechanics12% · ≈8 Qs

Atomic Physics10% · ≈6 Qs

Thermodynamics and Statistical Mechanics10% · ≈6 Qs

Optics and Wave Phenomena9% · ≈6 Qs

Specialized Topics9% · ≈6 Qs

Special Relativity6% · ≈4 Qs

Laboratory Methods6% · ≈4 Qs

Practice Questions by Topic

Use Start Test for a full weighted GRE Physics simulation, or open the hub and pick a single topic to drill your weak area. After each full exam, your results show a per-topic breakdown so you know exactly where to focus — most candidates need the most reps on Classical Mechanics and Electromagnetism, the two largest areas.

Who Takes the GRE Physics Test?

There are no formal eligibility requirements to take the GRE Physics Subject Test — anyone may register.^[3] In practice, the test assumes the equivalent of roughly three years of undergraduate physics coursework, so it is taken almost exclusively by students applying to graduate physics (and related) programs. Some programs require or recommend it while many have made it optional, so always confirm each program’s current requirements directly, since policies have shifted in recent years.

How Do You Register for the GRE Physics Test?

You register for the GRE Physics Subject Test online through your ETS account at ets.org. The GRE Subject Tests are now computer-delivered and offered during multiple administration windows each testing year at test centers worldwide (and at home where available).^[3] The current published fee is $150, which includes sending scores to up to four graduate institutions — verify the fee and any fee-reduction options before you register, as ETS pricing is subject to change.^[4]

How Is the GRE Physics Test Scored?

The GRE Physics Subject Test is scored on a scaled range of 200 to 990 in 10-point increments, with no pass/fail result. The total score is accompanied by a percentile rank showing the percentage of test takers who scored below you.

^[5] For tests taken since September 2023, ETS also reports three percent-correct subscores — Classical Mechanics, Electromagnetism, and Quantum Mechanics and Atomic Physics — on a 0–100 scale, plus an overall percent-correct figure.

Because admissions committees interpret scores relative to other applicants, the percentile rank matters as much as the scaled number.^[6]

How Hard Is the GRE Physics Test?

There is no pass/fail and therefore no pass rate — performance is reported as a scaled score and percentile rank, so you are evaluated relative to the cohort of test takers.^[6] The test is widely regarded as difficult, primarily because of its breadth and pace: it compresses roughly three years of undergraduate physics into about 100 questions in 170 minutes — well under two minutes per question — so it rewards quick recall, strong problem-solving, and fast elimination of answer choices.

200–990

Scaled score range

10-point increments

~700

Reported average total

aim above the median

Content areas

all tested

The takeaway: review the full breadth of the syllabus, drill timed practice across all nine areas, and memorize key constants and relationships so you can move fast on test day.

What to Expect on Exam Day

Arrive at your test center at least 15 minutes early to check in — bring a valid, unexpired government-issued photo ID whose name matches your ETS registration.^[3] You’ll store phones and personal items; no notes or personal calculators are allowed.

A short tutorial precedes the exam, then you have 2 hours 50 minutes to answer about 100 five-option multiple-choice questions spanning all nine content areas. Because there’s no guessing penalty, never leave a question blank, and budget your time so the breadth doesn’t outrun the clock.

If you test at home, expect a similar environment and ID scan. ETS reports your scaled score and percentile after scoring. Having simulated the full timing with practice tests makes that pace feel routine.

How to Use This GRE Physics Practice Test

Recreate exam conditions. Take the full test timed, with no notes.
Diagnose, then drill. Use a full simulation to find weak topics, then drill them.
Prioritize the big areas. Classical Mechanics and Electromagnetism are the biggest score-movers.
Learn the why. Read every explanation — understanding beats memorizing.
Answer everything. There’s no guessing penalty, so never leave a question blank.

Why Take the GRE Physics Test?

A strong GRE Physics score gives graduate physics programs a common yardstick to compare applicants from different undergraduate backgrounds, and a high percentile rank can strengthen an application to competitive programs.^[1] These free GRE Physics practice tests are the most efficient way to build that score.

Conclusion

A competitive GRE Physics score comes down to breadth, speed, and quick recall across all nine content areas. Use this free GRE Physics practice test to find your weak topics, drill them to mastery, and reinforce them with our study guide, flashcards so you walk in confident on test day.

GRE Physics Practice Test FAQ

The GRE Physics Subject Test is a standardized admissions exam administered by ETS (Educational Testing Service) and used by graduate physics programs to compare applicants. It covers material from about the first three years of undergraduate physics and is taken by students applying to graduate school in physics and related fields.

References

1.ETS. “GRE Physics Subject Test: Content and Structure.” ETS.org, 2026. ↑
2.ETS. “GRE Physics Test Practice Book (content percentages).” ETS.org. ↑
3.ETS. “GRE Subject Test Registration.” ETS.org. ↑
4.ETS. “GRE Subject Test Fees.” ETS.org. ↑
5.ETS. “Understanding Your GRE Subject Test Scores.” ETS.org. ↑
6.ETS. “GRE Subject Test Scores (scale and percentiles).” ETS.org. ↑

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