GRE Physics Reference Sheet

How to revise from this sheet

1. 1Cover the right-hand column and say the relationship out loud before reading it. Recognising a formula is not the same as recalling it, and only recall survives a timed section.
2. 2Work down a single domain at a time, then answer a few questions in that domain immediately. Recall that is never used decays fastest.
3. 3When an entry has a condition attached, rehearse the condition as part of the statement. Most exam traps are built on applying a correct formula outside its hypothesis.

ETS lists the GRE Physics Subject Test as computer-delivered with approximately 70 five-choice questions and a 2-hour testing time.

Constants, units, and measurement

Values to two or three figures, plus the estimation habits that convert a hard item into an elimination.

Mechanical and gravitational

g = 9.81 m/s^2; G = 6.674 x 10^-11 N m^2/kg^2; N_A = 6.022 x 10^23 /mol; R = 8.314 J/(mol K)

R = N_A k_B is worth recalling as a relation, not as two separate numbers.

Electromagnetic

c = 2.998 x 10^8 m/s; e = 1.602 x 10^-19 C; epsilon_0 = 8.854 x 10^-12 F/m; 1/(4 pi epsilon_0) = 8.99 x 10^9 N m^2/C^2; mu_0 = 4 pi x 10^-7 T m/A

c^2 = 1/(epsilon_0 mu_0) ties the three together and is a fast consistency check.

Quantum and atomic

h = 6.626 x 10^-34 J s; hbar = 1.055 x 10^-34 J s; k_B = 1.381 x 10^-23 J/K = 8.617 x 10^-5 eV/K; hc = 1240 eV nm; Bohr radius 0.529 x 10^-10 m; hydrogen ground state -13.6 eV

hc = 1240 eV nm converts a photon wavelength to an energy in one step and is the single most useful shortcut on the sheet.

Particle masses

electron 9.109 x 10^-31 kg = 0.511 MeV/c^2; proton 1.673 x 10^-27 kg = 938.3 MeV/c^2

Working in MeV/c^2 keeps relativistic energy arithmetic to one line.

Error propagation

$$\frac{\sigma_f}{|f|} = \sqrt{\left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2} \ \text{(product)}$$

for sums and differences the absolute uncertainties add in quadrature; for products and quotients the relative uncertainties do; a power n multiplies the relative uncertainty by |n|

Independent random errors are assumed; correlated errors do not combine this way.

Significant figures and estimation

a result carries no more precision than its least precise input; check the exponent and the unit before the digits

On a five-choice item an order-of-magnitude check often eliminates three options without any algebra.

Classical mechanics

The largest single domain. Decide between force, energy, and momentum before writing anything.

Newton's second law

$$\vec F_{\text{net}} = m\vec a = \frac{d\vec p}{dt}$$

net force = mass times acceleration; equivalently force = dp/dt

The momentum form is the one that survives variable mass and impulse questions.

Work, energy, power

$$W = \int \vec F \cdot d\vec r, \quad K = \tfrac{1}{2}mv^{2}, \quad P = \vec F \cdot \vec v$$

W = integral of F.dr; kinetic energy = mv^2/2; W_net = change in kinetic energy; P = F.v

Mechanical energy is conserved only when the non-conservative work is zero.

Momentum and collisions

$$\sum m_i \vec v_i = \text{constant}$$

total momentum is conserved with no net external force; elastic collisions conserve kinetic energy as well, perfectly inelastic ones leave the bodies with a common velocity

Momentum conservation holds in every collision; energy conservation does not. Choosing the wrong one is the classic error.

Rotation

$$\tau = I\alpha, \quad L = I\omega, \quad K_{\text{rot}} = \tfrac{1}{2}I\omega^{2}$$

torque = I alpha; angular momentum L = I omega; rotational kinetic energy = I omega^2/2; rolling without slipping gives v = omega r

Common moments of inertia: solid sphere 2MR^2/5, solid cylinder MR^2/2, hoop MR^2, rod about its centre ML^2/12. The parallel-axis theorem adds Md^2.

Simple harmonic motion

$$\omega = \sqrt{k/m}, \quad \omega = \sqrt{g/L}, \quad E = \tfrac{1}{2}kA^{2}$$

restoring force -kx gives angular frequency sqrt(k/m); a simple pendulum gives sqrt(g/L) for small amplitude; total energy is kA^2/2

The pendulum period is amplitude-independent only in the small-angle approximation.

Gravitation and orbits

$$U = -\frac{Gm_1m_2}{r}, \quad v_{\text{orb}} = \sqrt{\frac{GM}{r}}, \quad v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$$

F = G m1 m2 / r^2; potential energy -G m1 m2 / r; circular orbital speed sqrt(GM/r); escape speed sqrt(2GM/r)

Escape speed is the orbital speed times the square root of 2 — a relation worth recalling instead of two formulas.

Fluids

$$P + \rho gh + \tfrac{1}{2}\rho v^{2} = \text{constant}$$

buoyant force equals the weight of displaced fluid; continuity gives Av constant; Bernoulli gives P + rho g h + rho v^2/2 constant along a streamline

Bernoulli assumes steady, incompressible, non-viscous flow along one streamline.

Lagrangian shortcut

$$\frac{d}{dt}\frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q}$$

L = T - V, and d/dt of the derivative of L with respect to q-dot equals the derivative of L with respect to q

A coordinate absent from L gives a conserved conjugate momentum — often the fastest route to the answer.

Electromagnetism

Symmetry decides the method: Gauss for high symmetry, superposition otherwise.

Coulomb and field

$$F = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^{2}}, \quad V = \frac{kq}{r}$$

F = k q1 q2 / r^2 with k = 1/(4 pi epsilon_0); E = F/q; potential of a point charge is kq/r

Potential is a scalar, so superposing potentials is far cheaper than superposing fields.

Gauss's law

$$\oint \vec E \cdot d\vec A = \frac{q_{\text{enc}}}{\varepsilon_0}$$

the flux of E through a closed surface equals the enclosed charge divided by epsilon_0

Useful only with spherical, cylindrical, or planar symmetry; the field inside a conductor in equilibrium is zero.

Field from potential

$$\vec E = -\nabla V$$

E = -grad V; along one axis, E_x = -dV/dx

The minus sign is the trap; the field points from high to low potential.

Circuits

$$\tau_{RC} = RC, \quad \tau_{LR} = L/R, \quad P = I^{2}R = V^{2}/R$$

V = IR; series resistances add, parallel conductances add; series capacitances add reciprocally, parallel capacitances add; RC time constant is RC and LR is L/R

Capacitors and resistors combine by opposite rules — the most common circuit-arithmetic slip.

Kirchhoff's rules

$$\sum I_{\text{in}} = \sum I_{\text{out}}, \qquad \sum \Delta V_{\text{loop}} = 0$$

currents into a junction sum to zero; potential differences around a closed loop sum to zero

They express charge conservation and energy conservation; sign conventions must be fixed before writing the equations.

Magnetic force and sources

$$\vec F = q\vec v \times \vec B, \quad B = \frac{\mu_0 I}{2\pi r}, \quad B = \mu_0 n I$$

F = qv x B and F = I L x B; a long straight wire gives B = mu_0 I/(2 pi r); a solenoid gives B = mu_0 n I

The magnetic force does no work, because it is always perpendicular to the velocity.

Induction

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

the induced emf is minus the rate of change of magnetic flux; the induced current opposes the change that produced it

Lenz's law is the minus sign, and it is what most induction items actually test.

Maxwell's equations and waves

$$c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}, \qquad \frac{E}{B} = c$$

Gauss for E, no magnetic monopoles, Faraday's law, Ampere-Maxwell law; in vacuum c = 1/sqrt(epsilon_0 mu_0), and E/B = c with E, B, and propagation mutually perpendicular

The absence of magnetic monopoles is the equation students most often forget to state.

Quantum mechanics

Two solved systems and a handful of structural rules carry nearly every item.

Schrodinger equation

$$-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}} + V\psi = E\psi$$

the time-independent equation sets the sum of kinetic and potential terms acting on psi equal to E psi

Bound states require the wavefunction to be normalisable and continuous, with a continuous derivative wherever V is finite.

Infinite square well

$$E_n = \frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}$$

E_n = n^2 pi^2 hbar^2 / (2 m L^2) with n = 1, 2, 3, ...; the wavefunctions are sine functions vanishing at both walls

Energies scale as n^2 and as 1/L^2 — enough to answer most scaling questions without the constant.

Harmonic oscillator

$$E_n = \left(n + \tfrac{1}{2}\right)\hbar\omega$$

E_n = (n + 1/2) hbar omega with n = 0, 1, 2, ...

Equally spaced levels with a nonzero ground-state energy; the half is the zero-point energy.

Uncertainty principle

$$\Delta x\,\Delta p \ge \frac{\hbar}{2}, \qquad \Delta E\,\Delta t \ge \frac{\hbar}{2}$$

the product of position and momentum uncertainties is at least hbar/2; energy and time obey the same bound

It gives order-of-magnitude ground-state energies directly, which is often all a question needs.

de Broglie and photons

$$\lambda = \frac{h}{p}, \qquad E = hf = \frac{hc}{\lambda}$$

lambda = h/p for matter; photon energy E = hf = hc/lambda; the photoelectric threshold is the work function

With hc = 1240 eV nm, a 500 nm photon carries about 2.5 eV.

Operators and measurement

$$\langle A \rangle = \langle \psi | \hat A | \psi \rangle$$

observables are Hermitian operators with real eigenvalues; the probability of an outcome is the squared modulus of the amplitude; commuting observables share eigenstates

Two observables can be simultaneously definite exactly when their commutator vanishes.

Angular momentum and spin

$$L^{2} \to l(l+1)\hbar^{2}, \quad L_z \to m\hbar$$

L^2 has eigenvalue l(l+1)hbar^2 with l = 0,1,2,...; L_z has m hbar with m from -l to l; the electron has spin 1/2 with two states

The degeneracy of a level with quantum number l is 2l + 1 before spin is counted.

Thermodynamics and statistical mechanics

Track which quantity is held fixed; the process determines everything else.

First law

$$\Delta U = Q - W$$

the change in internal energy equals heat added minus work done by the system

Fix the sign convention once: W here is work done by the system.

Ideal gas

$$PV = nRT = Nk_BT, \qquad U = \tfrac{3}{2}nRT$$

PV = nRT = N k_B T; internal energy of a monatomic ideal gas is 3nRT/2

Internal energy of an ideal gas depends on temperature alone, which settles many process questions immediately.

Processes

$$W_{\text{iso}} = nRT\ln\frac{V_2}{V_1}, \qquad PV^{\gamma} = \text{constant}$$

isothermal: U unchanged and W = nRT ln(V2/V1); adiabatic: Q = 0 and PV^gamma constant; isochoric: W = 0; isobaric: W = P dV

gamma = C_P/C_V is 5/3 for a monatomic and 7/5 for a diatomic ideal gas at moderate temperature.

Equipartition and kinetic theory

$$v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}$$

each quadratic degree of freedom carries k_B T/2 per molecule; the rms speed is sqrt(3 k_B T/m)

Heavier molecules at the same temperature are slower by the inverse square root of mass.

Second law and entropy

$$dS = \frac{dQ_{\text{rev}}}{T}, \qquad S = k_B \ln W$$

entropy never decreases in an isolated system; for a reversible step dS = dQ/T; statistically S = k_B ln W

Entropy is a state function, so any convenient reversible path computes the change.

Heat engines

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

efficiency = W/Q_hot; the Carnot bound is 1 - T_cold/T_hot with absolute temperatures; a refrigerator's coefficient of performance is Q_cold/W

The temperatures must be in kelvin, and no engine between the same reservoirs can beat the Carnot value.

Distributions

Maxwell-Boltzmann for distinguishable classical particles; Fermi-Dirac for half-integer spin with at most one particle per state; Bose-Einstein for integer spin with no such limit

The exclusion principle is what makes metals and white dwarfs behave as they do; the statistics question is usually asking which of the three applies.

Optics and waves

Path difference for interference, aperture size for diffraction, sign conventions for lenses.

Wave basics

$$v = f\lambda, \qquad v = \sqrt{T/\mu}$$

v = f lambda; on a string v = sqrt(T/mu); the Doppler shift raises the observed frequency for approach and lowers it for recession

Frequency is set by the source and does not change when a wave enters a new medium; wavelength does.

Standing waves

$$\lambda_n = \frac{2L}{n} \quad \text{or} \quad \lambda_n = \frac{4L}{n},\ n \text{ odd}$$

a string fixed at both ends, or a pipe open at both ends, has wavelengths 2L/n; a pipe closed at one end has 4L/n with n odd only

The closed pipe supports only odd harmonics, which is why its timbre differs.

Two-slit interference

$$d\sin\theta = m\lambda$$

bright fringes where d sin(theta) = m lambda; dark fringes at half-integer multiples

Fringe spacing on a distant screen is about lambda L/d; wider slit separation means finer fringes.

Single-slit diffraction

$$a\sin\theta = m\lambda,\quad m = \pm 1, \pm 2, \dots$$

minima where a sin(theta) = m lambda for nonzero integer m; the central maximum is twice as wide as the others

Note the reversal: the same equation gives maxima for two slits and minima for one slit.

Thin films and gratings

a reflection off a higher-index medium adds a half-wavelength shift; a grating gives maxima at d sin(theta) = m lambda with sharper peaks as the number of lines grows

Count the phase shifts at both surfaces before deciding whether constructive interference needs an integer or half-integer path difference.

Geometric optics

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}, \qquad n_1\sin\theta_1 = n_2\sin\theta_2$$

1/f = 1/d_o + 1/d_i; magnification m = -d_i/d_o; Snell's law n1 sin(theta1) = n2 sin(theta2); total internal reflection above the critical angle sin(theta_c) = n2/n1

Converging lenses and concave mirrors have positive f; a negative image distance means a virtual image.

Atomic, nuclear, and condensed matter

Scaling laws and selection rules; the numbers to hold are few and the patterns do the rest.

Hydrogen spectrum

$$E_n = -\frac{13.6\ \text{eV}}{n^{2}}$$

E_n = -13.6 eV / n^2; a transition emits a photon of energy equal to the level difference

For a hydrogen-like ion of charge Z the energies scale as Z^2, so helium-plus has -54.4 eV in its ground state.

Atomic structure rules

the Pauli exclusion principle forbids two electrons in the same quantum state; Hund's rule maximises spin in a partly filled subshell; electric dipole transitions require the orbital quantum number to change by one

Selection rules explain which spectral lines are absent, a standard multiple-choice hook.

Nuclear decay

$$N(t) = N_0e^{-\lambda t}, \qquad t_{1/2} = \frac{\ln 2}{\lambda}$$

N = N_0 exp(-lambda t); the half-life is ln(2)/lambda; alpha decay reduces mass number by 4 and charge by 2, beta-minus raises charge by 1 at constant mass number

Binding energy per nucleon peaks near iron, which is why both fusion of light nuclei and fission of heavy ones release energy.

Blackbody radiation

$$P/A = \sigma T^{4}, \qquad \lambda_{\max}T = 2.90 \times 10^{-3}\ \text{m K}$$

the emitted power per area is sigma T^4 with sigma = 5.67 x 10^-8 W/(m^2 K^4); the peak wavelength is inversely proportional to temperature

The fourth power means a factor of 2 in temperature is a factor of 16 in radiated power.

Solids

$$2d\sin\theta = m\lambda$$

conductors have a partly filled band, semiconductors a small gap, insulators a large one; the lattice spacing follows from Bragg's law 2 d sin(theta) = m lambda

Bragg's law uses the angle from the crystal plane, not from the normal — the opposite convention to optics.

Special relativity

Fix the frame first, then apply one relation. Most errors are frame errors, not algebra errors.

Lorentz factor

$$\gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}}$$

gamma = 1/sqrt(1 - v^2/c^2), always at least 1

Every relativistic relation is a factor of gamma applied to a classical one; knowing where it goes matters more than the algebra.

Time dilation and length contraction

$$\Delta t = \gamma \Delta t_{\text{proper}}, \qquad L = \frac{L_{\text{proper}}}{\gamma}$$

a moving clock runs slow by gamma; a moving length contracts by gamma along the direction of motion only

Proper time is measured where the two events happen at the same place; proper length where the object is at rest.

Velocity addition

$$u' = \frac{u - v}{1 - uv/c^{2}}$$

u' = (u - v)/(1 - uv/c^2), which never exceeds c

Setting u = c returns c in every frame — a fast check that the formula has been applied correctly.

Energy and momentum

$$E^{2} = (pc)^{2} + (mc^{2})^{2}$$

E = gamma m c^2; p = gamma m v; E^2 = (pc)^2 + (mc^2)^2; kinetic energy is (gamma - 1) m c^2

The invariant relation is the workhorse: for a photon m = 0 so E = pc, and at low speed it reduces to the classical kinetic energy.

Invariants

$$s^{2} = (c\Delta t)^{2} - \Delta x^{2}$$

the spacetime interval and the rest mass are the same in every inertial frame

When a problem looks frame-dependent, compute an invariant instead — it removes the frame from the question.

Sources

Entries drawn from a cited source list it below. Standard results and numerical constants that no source in our registry covers are given as recall material without a citation rather than attributed to a source that does not support them.

1. GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
2. University Physics Volume 1, Section 5.3: Newton's Second Law — OpenStax. Accessed 2026-07-06. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
3. University Physics Volume 1, Section 8.3: Conservation of Energy — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
4. University Physics Volume 1, Section 9.3: Conservation of Linear Momentum — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
5. University Physics Volume 1, Section 15.1: Simple Harmonic Motion — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
6. University Physics Volume 2, Section 6.3: Applying Gauss's Law — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
7. University Physics Volume 2, Section 10.3: Kirchhoff's Rules — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
8. University Physics Volume 3, Section 7.2: The Heisenberg Uncertainty Principle — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
9. University Physics Volume 3, Section 7.4: The Quantum Particle in a Box — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
10. University Physics Volume 2, Section 3.3: First Law of Thermodynamics — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
11. University Physics Volume 2, Section 4.5: The Carnot Cycle — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
12. University Physics Volume 3, Section 3.1: Young's Double-Slit Interference — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
13. University Physics Volume 3, Section 4.1: Single-Slit Diffraction — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
14. University Physics Volume 3, Section 5.3: Time Dilation — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
15. University Physics Volume 3, Section 5.9: Relativistic Energy — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
16. University Physics Volume 1, Section 1.6: Significant Figures — OpenStax. Accessed 2026-08-15. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.