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• Taylor Faucett

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Physics GRE Equations Sheet Version 1.0 Contact: [email protected]

Contents 1 Classical Mechanics

1

2 Electricity & Magnetism

3

3 Optics

4

4 Thermodynamics & Statistical Mechanics

5

5 Quantum Mechanics

7

6 Special Relativity

8

7 Electronics

9

8 Special Topics

1

10

Classical Mechanics

Equations Kinematics: 1 ∆x = v0 t + at2 2 v 2 = v02 + 2ad v = +at v0 + vf ∆x = t 2 Newton’s Second Law: F = ma Momentum: P = mv Centripetal Acceleration: ac = ω 2 r = mrω 2 2π ω = T

F

1

1

Gravitational Potential Energy: U U U

= mgh m1 m2 = −G R = −W

Kinetic Energy: 1 mv 2 2 p2 2m 1 2 Iω 2

Uk = Uk = Uk−rotational = Parallel Axis Theorem:

I = Icom + M h2 Work: Z

F ·s

W = Moment of Inertia (Point Mass):

I = M R2 Schwarzschild Radius: Re =

2GM c2

Lagrangian: L=T −V Hamiltonian: H =T +V Angular Frequency: ω =

2π T s

k m

ωSHO = Period of a SHO: r

T = 2π

2

m k

CLASSICAL MECHANICS

2

2

Electricity & Magnetism

Equations Maxwell’s Equations: ρ 0 ~ ∇·B = 0 ~ ~ = − ∂B ∇×E ∂t ~ = ∇·E

~ ~ = µ0 J~ + µ0 0 ∂ E ∇×B ∂t Coulombs Law: U

=

U

=

F

=

1 dQ 4π0 R 1 q1 q2 4π0 R 1 q1 q2 4π0 R2

Hooke’s Law: F = −kx Faraday’s Law of Induction dφB |ε| = N dt

Magnetic Flux: Z Z

~ r, t) · dA B(~

φB = S

Malus’ Law: 1 I = c0 E02 cos2 (θ) 2 Gauss’ Law: ~ · dA ~= q E 0 Larmor Formula: P =

q 2 a2 6π0 c3

Lorentz Force: 

~ F = q ~v × B Hall Voltage: VH = −

3

IB dne



ELECTRICITY & MAGNETISM

3

3

Optics

Equations Photoelectric Equation: Uk = hν − φ Snell’s Law: n2 sin(θ1 ) = sin(θ2 ) n1 n-slit constructive interference: d sin(θn ) = nλ n-slit destructive interference: 1 λ d sin(θn ) = n + 2 



Photon Energy: E = hν hc E = λ Interference of a thin film: 

1 λ 2 

2nd = m + Light speed through a medium: v=

1 µ

Traveling wave: 2π y(x, t) = A sin (x ± vt) λ 



Drift velocity: vd =

i nqA

Compton Equation: λ0 − λ =

h (1 − cos(θ)) mp c

4

OPTICS

4

4

THERMODYNAMICS & STATISTICAL MECHANICS

Thermodynamics & Statistical Mechanics

Equations Rydberg Formula: 1 1 1 − 2 = R∞ 2 λ n 1 n2 



Rydberg Formula for Hydrogen like atoms: 1 λH−like

= RZ

2



1 1 − 2 2 n1 n2



Moseley’s Law: 1 λK−α

= R (Z − β)

2



1 1 − 2 2 n 1 n2



Rydberg Energy: me e4 Z 2 2 n2 2¯ h  1 1 E = E0 − λ21 λ2   1 1 E = n(13.6 eV ) − λ21 λ22 E = −

Heat Capacity: ∆Q ∆T ∂Q ∂T ∂S T  ∂T    ∂Q ∂U = ∂T V ∂T V     ∂Q ∂H = ∂T p ∂T p

C = C = C = CV

=

Cp = Heat:

Q = cm∆T First Law of Thermodynamics: dU

= dQ + dW

dU

= dQ − P dV

Ideal Gas Law: pV = nRT Thermodynamic Work: Z Vf

W =

P dV Vi

5

4

THERMODYNAMICS & STATISTICAL MECHANICS

Entropy: ∆S =

Z T2 dq T1

T

Fourier’s Law of Heat Conduction: ∂Q = −k ∂t

I

~ ∇T · dA

s

Mean free path: P (x) = nσdx Stefan-Boltzmann’s Law: j ∗ = σT 4

6

5

5

Quantum Mechanics

Equations Particle Location (Probability): Z b

Pab =

|ψ(x)|2 dx

a

Infinite Square Well/Particle in a box: ψn (x, t) = Asin(kn x)e−iωn t Planck Length: s

lp =

G¯h c3

Expectation Value: hAi =

Z

hψ |A| ψi

Heisenberg Uncertainty Principle: ∆x∆p ≥

¯ h 2

Spin Operator: S12 ψ1 = S1 (S1 + 1)ψ1 Probability Current: ¯ ∂ψ ∂ψ ∗ ~ t) = h J(x, ψ∗ − ψ 2mi ∂x ∂x 



Quantum Harmonic Oscillator 

En = h ¯ω n +

1 2



Wave Speed (de Broglie relations): vp = vp =

E p c2 v

Time-Independent Schrodinger Equation: E ψ(x) = −

¯2 2 h ∇ ψ(x) + V (x) ψ(x) 2m

7

QUANTUM MECHANICS

6

6

Special Relativity

Equations Rest Energy: E = m0 c2 Lorentz Factor: 1 γ=q 1−

v2 c2

Relativistic Energy: ER = γmc2 Relativistic Momentum: m0 v prel = γm0 v = q 2 1 − vc2 Relativistic Energy-Momentum: 

E 2 = mc2

2

+ (pc)2

Relativistic sum of velocities: u0 = u =

u+v 1 + vu c2 u0 + v 0 0 0 1 + vcu2

Proper Time: ∆τ 2 = ∆t2 − ∆x2 ∆t2 = ∆τ 2 + ∆x2 Space-Time Interval: ∆S 2 = −(C∆t)2 + ∆x2

8

SPECIAL RELATIVITY

7

7

Electronics

Equations Ohm’s Law: V = IR Kirchoff ’s First Law: n X

I=0

k=1

Kirchoff ’s Second Law: n X

V =0

k=1

Current: i=

dq dt

Faraday’s law of induction: dφB dt

ε = Capacitance

C=

Q V

Frequency of an RLC Circuit f=

9

1 √ 4π LC

ELECTRONICS

8

8

Special Topics

Equations Acoustic Beats: beats = |f2 − f1 | Doppler Effect: "

f=

1 1±

10

vs ource vw ave

#

f0

SPECIAL TOPICS