Physics 251 Final Formula Sheet - Web Physics

Physics 251 Final Formula Sheet

Thermal Expansion ∆L = αL0 ∆T

†

indicates formulas that are specific to ideal gases. Cyclic Processes

β = 3α W =Q Q = mc∆T Isochoric Processes

Q = ±mL Q = nCV ∆T

W =0

Q = nCP ∆T

Q = nCV ∆T †

Heat Current A (TH − TC ) L Q = AeσT 4

H=k

∆U = nCV ∆T

Isobaric Processes W = p∆V Q = nCP ∆T

Heat Engines †

W = QH + QC W e= QH TC ecarnot = 1 − TH Refrigerators W = QH + QC |QC | K= |W | TC = TH − TC

Kcarnot

∆U = nCV ∆T

Isothermal Processes ∆U = 0   Vf † W = nRT ln Vi   V f † Q = nRT ln Vi

V2

V1

Ideal gases pV = nRT pV = N kT r 3RT vrms = m CP = CV + R CP γ= CV V √ λ = vtmean = 4π 2r2 N

Collection of Charges

~T = E

X

UT =

X

~i E

i

Q=0

Uij

pairs

∆U = W †

†

pdV

Z

dq rˆ 4π0 r2 Z dq V = 4π0 r ~ = E

Adiabatic Proceses

†

W =

Distributed Charges

†

Work Z

q rˆ 4π0 r2 q V = 4π0 r q1 q2 rˆ F~ = 4π0 r2 q1 q2 U= 4π0 r ~ = E

∆U = 0

Heat

Point Charges

W = −nCV ∆T

Electric Dipoles

P V γ = constant

T V γ−1 = constant ~ = pE sin φ ~τ = p~ × E ~ = −pE cos φ U = −~ p·E

Entropy Z ∆S = 1

2

dQ T

Electric Fields

“Elementary” E fields

~ F~ = q E ∆U = q∆V Wa→b = Ua − Ub = −∆U Z b ~ · d~l Vab = E a

~ = −∇V E 1

Q rˆ sphere 4π0 r2 ~ = λ rˆ line E 2π0 r σ ~ E= n ˆ plane 20

~ = E

Capacitors and Dielectrics

RC circuits

Q V X parallel: Ceq = Ci C=

τ = RC

A Parallel-plate: C = 0 d 2 Q 1 1 = QV U = CV 2 = 2 2C 2 1 u = 0 E 2 2 dielectric: C = KC0  = K0 Current & Current Density I=

t

I = I0 e− τ   t Q = Qf 1 − e− τ

t

VL = V0 e− τ

I = I0 e

− τt

Q = Q0 e

t

VL = V0 e− τ

~ F~ = q~v × B ~ dF~ = Id~l × B ~ ~τ = µ ~ ×B

mv qB 2πm T = qB qB ωc = m

Ohm’s Law ~ = ρJ~ E V = IR

Magnetic Fields dL R= ρ A ρ = ρ0 [1 + α(T − T0 )] Uniform currents V L L R=ρ A I |J| = A

|E| =

~ = µ0 q~v × rˆ B 4π r2 Z ~ ~ = µ0 Idl × rˆ B 4π r2 µ0 I |B| = ∞ straight wire 2πr µ0 I |B| = center of loop 2R |B| = µ0 nI ∞ solenoid µ0 N I |B| = toroidal solenoid 2πr Mutual Inductance

Electric Power M= P = IV P = I 2R 2

V P = R Resistors

XL = ωL 1 XC = pωC Z = R2 + (XL − XC )2 XL − XC tan φ = R I = I0 cos ωt

N 2 Φ2 N 1 Φ1 = I2 I1 dI2 E1 = −M dt dI1 E2 = −M dt

V0 = I0 Z V0 Vrms = √ 2 I0 Irms = √ 2 1 2 1 P = I0 R = I02 Z cos ϕ 2 2 AC circuits (resonance)

ϕ=0 Z=R 1 LC XL = XC

ω=√

Maxwell’s Equations I ΦE = S I

ΦB =

Self Inductance

series: Req =

X

Ri

i

parallel:

AC circuits

V = V0 cos(ωt + ϕ)

Resistivity & Resistance Z

t

I = I0 e− τ

Magnetic Force

R=

J = nqvd

discharging:

− τt

Cyclotron Motion

J~ · d~a

I=

  t I = If 1 − e− τ

discharging:

~ U = −~ µ·B

dq ZdtZ

L R charging: τ=

charging:

i

X 1 1 = serial: Ceq Ci i

RL circuits

X 1 1 = Req Ri i

real battery: V = E − Ir

~ · dA ~ = Qencl E 0 ~ · dA ~=0 B

S

NΦ I dI E = −L dt 1 2 U = LI 2

I

L=

~ · d~l = µ0 (ic + 0 dΦE )encl B dt

C

I C

2

~ · d~l = −0 dΦB = E E dt

Light

Emax = cBmax 1 c= √ 0 µ0 c = λf ~= 1E ~ ×B ~ S µ0 I ~ · dA ~ P = S I = Sav prad prad

Emax Bmax = 2µ0 Sav absorbed = c 2Sav = reflected c

Light Propagation c n= v λ0 λ= n na sin θa = nb sin θb nb sin θc = (T IR) na I = I0 cos2 ϕ Mirrors and Lenses f=

R 2

Constants J mol · K NA = 6.02 × 1023 R J kB = = 1.38 × 10−23 NA mol · K qe = e = −1.602 × 10−19 C R = 8.3145

k=

1 N · m2 = 8.99 × 109 4π0 C2 8 c = 3.00 × 10 m/s me = 9.11 × 10−31 kg

1 1 1 = + 0 f s s   1 1 1 = (n − 1) − f R1 R2 y0 s0 m= =− y s

3

mp = 1.67 × 10−27 kg