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Cheat Sheet — Midterm 2 PHYS 142 (Summer 2014) Constants: ǫ0 = 8.9 × 10−12 C2 /N · m2 e = 1.6 × 10−19 C

g = 9.8 m/s2

µ0 = 4π × 10−7 Wb/A · m

me = 9.1 × 10−31 kg

c = 3.0 × 108 m/s

mp = 1.7 × 10−27 kg

Charges, Currents, and the Electromagnetic Field Forces on Charges and Currents: ˆ ~ = q( E ~ +v ~ ~ = Id~l × B ~ ~ × B) F F

Flux: ΦE =

¨

A

~ ·n E ˆ dA

ΦB =

¨

A

~ ·n B ˆ dA

Maxwell’s Equations: Qenc ~ ·n E ˆ dA = ǫ0 A

‹

‹

A

~ · d~l = − dΦB E dt l

~ · d~l = µ0 Ienc + µo ǫ0 dΦE B dt l

˛

~ ·n B ˆ dA = 0

˛

Sources of Electric and Magnetic Field: ~ = E

1 q rˆ 4πǫ0 r 2

Electrostatic Potential: ˆ 1 dq V = r 4πǫ0

~ = E

1 4πǫ0

ˆ

Va − Vb =

ˆ

v × ˆr ~ = µ0 q~ B 4π r 2

dq ˆr r2

b a

l

Id~l × rˆ r2

Magnetic Dipoles:

~ τ~ = ~p × E

Energy in Fields: U=

ˆ

~ = − ∂V ˆı − ∂V ˆ − ∂V ˆk ~ = −∇V E ∂z ∂x ∂y

~ · d~l E

Electric Dipoles ~ p = q~d

~ = µ0 B 4π

~ U = −~p · E

µ ~ = IAˆ n

Motion in B-Field:

1 1 2 B ǫ0 E 2 + 2µ 2 0

R=

~ ~τ = ~µ × B

~ U = −~µ · B

B-Fields of Infinite Objects:

mv |q|B

B=

µ0 I (wire) 2πr

B = µ0 nI (soleniod)

Electrical Circuits Current: I=

dQ = n|q|vd A dt

Kirchoff’s Rules: ~ = nq~vd J

ρ=

E J

X

V =0

loop

Faraday’s Law: X

I =0

junction

E = −N

Capacitors: Q C= V

n X 1 1 (Series) = C Ceq i i=1

Ceq =

n X i=1

Ci (Parallel)

U=

Q2 1 CV 2 = 2 2C

dΦB dt

Resistors: V = IR

n X 1 1 (Parallel) = R Req i=1 i

Req =

n X

di dt

L=

V2 P = I 2R = R

Mutual Indcutors:

N ΦB i

1 U = Li2 2

E1 = −M

di2 dt

E2 = −M

di1 dt

M=

N2 ΦB2 N1 ΦB1 = i2 ii

R-C Circuit (discharging):

q = CE(1 − e−t/RC )

i=

ρL A

R-C Circuit (charging):

R-L Circuit: L τ= R

R=

i=1

Inductors (single): E = −L

Ri (Series)

i=

E −t/RC e R

q = Q0 e−t/RC

E (1 − e−(R/L)t) (current growth) R

L-C Circuit: r 1 q = Q cos(ωt + φ) ω= LC AC Circuits: I V Irms = √ Vrms = √ 2 2

i=−

Q0 −t/RC e RC

i = I0 e−(R/L)t (current decay)

R-L-C Circuit (underdamped): r R2 1 ′ ω = q = Ae−(R/2L)t cos(ω ′ t + φ) − 2 LC 4L

i = Icosωt

L-R-C Circuit: p V = IZ Z = R2 + (XL − XC )2

v = V cos(ωt+φ)

tan φ =

VR = IR

VL = IXl

VC = IXC

Power in AC Circuits: 1 Pav = V I cos φ = Vrms Irms cos φ 2

XL − XC R

Transformers: V2 N2 = V1 N1

V1 I1 = V2 I2

Electromagnetic Waves and Light Waves in Vacuum: E = cB

Energy/Momentum:

1 c= √ µ 0 ǫ0

~ B ~S = 1 E× ~ µ0

c = fλ

Index of Refraction: prad =

S 1 dp = C A dt

n=

c λ0 = v λ

Sinusoidal EM Waves (in +x direction): ˆ max cos(kx − ωt) ~ = kB B

~ E = ˆEmax cos(kx − ωt)

Emax = cBmax

Reflection/Refraction: θa = θr

I = Sav =

EmaxBmax 2µ0

Polariziation:

na sin θa = nb θb

sin θcrit =

nb na

I = I0 cos2 φ

tan θp =

nb na

Two-source Interference: d sin θ = mλ (con.)

 1 d sin θ = m+ λ (des.) 2

ym = R

Single Slit Diffraction: sin θ =

mλ a

I = I0

 sin[πa(sin θ)/λ] 2 πa(sin θ)/λ

2

mλ d

 φ φ  EP = 2E cos  I = I0 cos2 2 2

φ = k(r2 −r1 )

Diffraction Grating:

X-ray Diffraction:

d sin θ = mλ

2d sin θ = mλ...