Cheat Sheet — Midterm 2

PHYS 142 (Summer 2014)

Constants:

ε0 = 8.9× 10−12 C2/N ·m2 µ0 = 4π × 10−7 Wb/A ·m g = 9.8 m/s2

e = 1.6× 10−19 C me = 9.1× 10−31 kg mp = 1.7× 10−27 kg c = 3.0× 108 m/s

Charges, Currents, and the Electromagnetic Field

Forces on Charges and Currents: Flux:

~F = q(~E + ~v × ~B) ~F =

ˆId~l× ~B ΦE =

¨A

~E · n dA ΦB =

¨A

~B · n dA

Maxwell’s Equations:‹

A

~E · n dA =Qenc

ε0

‹A

~B · n dA = 0

˛l

~E · d~l = −dΦB

dt

˛l

~B · d~l = µ0Ienc + µoε0dΦE

dt

Sources of Electric and Magnetic Field:

~E =1

4πε0

q

r2r ~E =

1

4πε0

ˆdq

r2r ~B =

µ04π

q~v × r

r2~B =

µ04π

ˆl

Id~l× r

r2

Electrostatic Potential:

V =1

4πε0

ˆdq

rVa − Vb =

ˆ b

a

~E · d~l ~E = −~∇V = −∂V∂x

ı− ∂V

∂y− ∂V

∂zk

Electric Dipoles Magnetic Dipoles:

~p = q~d ~τ = ~p× ~E U = −~p · ~E ~µ = IAn ~τ = ~µ× ~B U = −~µ · ~B

Energy in Fields: Motion in B-Field: B-Fields of Infinite Objects:

U =1

2ε0E

2 +1

2µ0B2 R =

mv

|q|BB =

µ0I

2πr(wire) B = µ0nI (soleniod)

Electrical Circuits

Current: Kirchoff’s Rules: Faraday’s Law:

I =dQ

dt= n|q|vdA ~J = nq~vd ρ =

E

J

∑loop

V = 0∑

junction

I = 0 E = −N dΦB

dt

Capacitors:

C =Q

V

1

Ceq=

n∑i=1

1

Ci(Series) Ceq =

n∑i=1

Ci (Parallel) U =1

2CV 2 =

Q2

2C

Resistors:

V = IR1

Req=

n∑i=1

1

Ri(Parallel) Req =

n∑i=1

Ri (Series) R =ρL

AP = I2R =

V 2

R

Inductors (single): Mutual Indcutors:

E = −Ldidt

L =NΦB

iU =

1

2Li2 E1 = −Mdi2

dtE2 = −Mdi1

dtM =

N2ΦB2

ii=N1ΦB1

i2

R-C Circuit (charging): R-C Circuit (discharging):

q = CE(1− e−t/RC) i =ERe−t/RC q = Q0e

−t/RC i = − Q0

RCe−t/RC

R-L Circuit:

τ =L

Ri =ER

(1− e−(R/L)t) (current growth) i = I0e−(R/L)t (current decay)

L-C Circuit: R-L-C Circuit (underdamped):

ω =

√1

LCq = Q cos(ωt+ φ) ω′ =

√1

LC− R2

4L2q = Ae−(R/2L)t cos(ω′t+ φ)

AC Circuits:

Irms =I√2

Vrms =V√

2i = Icosωt v = V cos(ωt+φ) VR = IR VL = IXl VC = IXC

L-R-C Circuit: Power in AC Circuits:

V = IZ Z =√R2 + (XL −XC)2 tanφ =

XL −XC

RPav =

1

2V I cosφ = VrmsIrms cosφ

Transformers:V2V1

=N2

N1V1I1 = V2I2

Electromagnetic Waves and Light

Waves in Vacuum: Energy/Momentum: Index of Refraction:

E = cB c =1

√µ0ε0

c = fλ ~S =1

µ0~E×~B prad =

1

A

dp

dt=S

Cn =

c

v=λ0λ

Sinusoidal EM Waves (in +x direction):

~E = Emax cos(kx− ωt) ~B = kBmax cos(kx− ωt) Emax = cBmax I = Sav =EmaxBmax

2µ0

Reflection/Refraction: Polariziation:

θa = θr na sin θa = nbθb sin θcrit =nbna

I = I0 cos2 φ tan θp =nbna

Two-source Interference:

d sin θ = mλ (con.) d sin θ =(m+

1

2

)λ (des.) ym = R

mλ

dEP = 2E

∣∣∣ cosφ

2

∣∣∣ I = I0 cos2φ

2φ = k(r2−r1)

Single Slit Diffraction: Diffraction Grating: X-ray Diffraction:

sin θ =mλ

aI = I0

(sin[πa(sin θ)/λ]

πa(sin θ)/λ

)2d sin θ = mλ 2d sin θ = mλ

2