PGRE Eqs

Newton’s second law

F = ma

Momentum

p = mv

Reduced mass of a two-body system

μ = m₁m₂/(m₁ + m₂)

Kinetic energy

K = 1/2(mv²) = p²/(2m)

Work

W = integral(F· dr)

Newton’s universal law of gravitation

F = Gm₁m₂/r²

Torque

τ = Iα

Angular momentum

L = r x p

Moment of Inertia (parallel axis theorem)

I = I_cm + md^²

Rotational Kinetic Energy

K = 1/2(Iω²)

Centripetal Acceleration

a = v²/r

Period (uniform circular motion)

T = 2πr/v

Angular velocity

ω = 2π/T = v/r

Cyclotron angular frequency of a charged particle moving in a magnetic field

ω = qB/m

Period (simple pendulum)

T = 2πsqrt(L/g)

Spring force

F = −k∆x

Potential energy of a stretched/compressed spring

U = 1/2k∆x²

Electric potential (point charge)

V(r) = q/4πε₀r

Electric Field/Electric Potential relation

E(r) = −∇V(r)

Force on a charged particle in a uniform E-field

F = qE

Gauss’ Law

surface integral(E · da) = 1/ε₀ volume integral(ρdV)

Force on a charged particle in a uniform magnetic field

F = qv × B

Biot-Savart Law

dB = (μ₀I/4π) integral((dℓ’ x (r - r’))/|r - r’|³

Ampere’s Law (integral form)

loop integral(B · dl) = μ₀ surface integral(J · da)

Faraday’s Law of induction

loop integral(E · dl) = -(d/dt) surface integral(B · da)

solenoid magnetic flux

Φ = NAB

Faraday’s Law (solenoid)

EMF = -dΦ/dt

Maxwell’s equations

∇ x H = J + ∂D/∂t

∇ x E = -∂B/∂t

∇ · D = ρ

∇ · B = 0

Poynting Vector

S = E x H

relationship between magnetic permeability, vacuum permeability, and speed of light

μ₀ ε₀ = 1/c²

Ohm’s Law (alternating current)

U = ZI

Impedance of an RLC circuit

Z = R + iωL + 1/iωC

Impedance (in series)

Z = Z1 + Z2 + . . .

Impedance (in parallel)

1/Z = 1/Z₁ + 1/Z₂ + . . .

Power (AC circuit)

P (t) = Re(I(t)) Im(V (t))

Doppler effect (sound)

f = ((c ± v_r) / (c ± v_s)) f₀

Index of refraction

n = c/v

Snell’s law

n₁sinθ₁ = n₂sinθ₂

Focal length of a refracting telescope

l = f_objective + f_eyepiece

Magnification of a refracting telescope

M = f_objective/f_eyepiece

Work done on an ideal gas during a volume change

W = −integral from V_i to V_f (PdV)

COP (efficiency) of a refrigerator

COP ≤ T_c / (T_h - T_c)

Maxwell-Boltzmann Speed Distribution Function

D(v) = (m/2πk_BT)^3/24πv²e^-mv²/2k_BT

Energy of a photon

E = hf

momentum operator

p = -ihbar∇

position-momentum commutation relation

[x,p] = ihbar

angular momentum magnitude squared-angular momentum vector commutation relation

[J², J] = 0

angular momentum x and y components commutation relation

[J_x, J_y] = ihbarJ_z

Pauli matrices product identity

σ_jσ_k = δ_jk + i ϵ_jkl σ_l

Infinite square well energy

E_n = n²π²hbar²/2ma²

harmonic oscillator energy

E_n = (n+1/2)hbarω

Characteristic energies (hydrogen-like atoms)

E_n = -(m_e(Ze²)²/2(4πε₀)²hbar²)(1/n²)

Allowed quantum numbers (hydrogen-like atoms)

n = 1, 2, 3, …

l = 0, 1, … (n-1)

m = -l, … +l

β (special relativity)

β = v/c

Lorentz factor (special relativity)

γ = 1/sqrt(1-β²)

momentum of a relativistic particle

p = γm₀v

Energy-momentum relation (special relativity)

E² = (pc)² + (m₀c²)²

length contraction

l’ = l₀/γ

time dilation

∆t′ = γ∆t₀

relativistic Doppler effect

f = sqrt((1 ± β)/ (1 ∓ β)) f₀

Poisson uncertainty

σ = sqrt(N)

error propagation

σ²_X = sum over i[(∂X/∂z_i)²σ²_zi]

Lagrangian

L = T - U

Hamilton’s principle

partial L / partial x - d/dt partial L / partial xdot = 0

Hamiltonian

H = sum of p_iqdot_i - L

Hamilton’s Equations

partial H / partial p_i = qdot_i

partial H / partial qdot_i = -pdot_i

force of friction

f = muN

Energy stored on a capacitor

E = ½ CV²

image location (mirrors)

1/image distance = 1/focal length - 1/object distance

image location (lenses)

1/ focal length = 1/object distance + 1/image distance

Rayleigh criterion

theta = 1.22 lambda/D