Equation Sheet

Constants ˙Coulomb’s law k = 8.99·109 Nm2/C2

= 1

4πǫ0Permittivity ǫ0 = 8.85·10−12 C2/Nm2

Permeability µ0 = 4π·10−7 Tm/AElectron charge −e=−1.60·10−19 CSpeed of light c = 3.8·108 m/s

Forces ˙

Coulomb: kq1q2

r2q ~E

Lorentz: q~v × ~B l~I × ~B

Centripetal:mv2

r

Fields .

point charge E = kQ

r2

infinitesimal Coulomb d ~E = kdQ

r2r

given a potential Ex = −∂V

∂x

infinite straight current B =µ0

2π

I

rsolenoid B = KBµ0In, n = N/l

infinitesimal Biot-Savart d ~B =µ0I

4π

d~l × r

r2

Flux

in general ΦX =∮

~X · d ~A

uniform ~X ΦX = ~X · ~Aoutflux ΦX > 0influx ΦX < 0

Maxwell’s Equations

(Gauss)

∮

~E · d ~A = Qenc/ǫ0

∮

~B · d ~A = 0

(Faraday)

∮

~E · d~l = −∂ΦB

∂t(= E)

(Ampere)

∮

~B · d~l = µ0Ienc + µ0ǫ0

∂ΦE

∂t

Electric Potential

point charge kq

rinfinitesimals dV =

∫

k dQr

given a field Vab = −∫ b

a~E · d~r

Potential Energy

point charges kq1q2

rpoint charge and field qV

capacitor 1

2QV = 1

2CV 2 = Q2

2Cinductor 1

2LI2

Miscellaneoustransformer Vs

Vp= Ns

Np

magnetic moment of current loops ~µ = NI ~A

torque ~τ = ~r × ~F

torque on a magnet ~τ = ~µ × ~Bparallel plate capacitance C = Kǫ0

Ad

RC circuit Q = Q0e−t/τ , τ = RC

Circuitscurrent I ≡ ∆Q/∆tcapacitance C ≡ Q/Vinductance L = NΦB

IM21 = N2Φ21

I1(Ohm) V = IRpower in resistor P = IV = I2R = V 2/Rseries capapacitors C−1

eq =∑

i C−1

i V =∑

i Vi Q = Qi

parallel capacitors Ceq =∑

i Ci V = Vi Q =∑

i Qi

series resistors Req =∑

i Ri V =∑

i Vi I = Ii

parallel resistors R−1

eq =∑

i R−1

i V = Vi I =∑

i Ii

series inductors Leq =∑

i Li V =∑

i Vi I = Ii

parallel inductors L−1

eq =∑

i L−1

i V = Vi I =∑

i Ii

(Kirchhoff) junction The sum of all currents entering the junction must equal the sumof all currents leaving the junction (charge is conserved).

(Kirchhoff) loop The sum of the changes in the potential around any closed path ofa circuit must be zero.

AC Circuitscurrent I(t) = I0 cos(2πft)voltage V (t) = V0 cos(2πft + φ)phase angle tan φ = (XL − XC)/R

RMS Irms = I0/√

2 Vrms = V0/√

2

impedance Z =√

R2 + (XL − XC)2 Vrms = IrmsZpower factor cosφ

resonance φ = 0 f0 = 1/(2π√

LC)power dissipated P = I2

rmsR

R V = IR Vrms = IrmsR I, V in phase φ = 0C dV

dt= I

CVrms = IrmsXC XC = (2πfC)−1 V lags I φ = −90o

L V = LdIdt

Vrms = IrmsXL XL = 2πfL I lags V φ = 90o

displacement current ID = ǫ0AdEdt

wave equation v = fλ

Pointing Vector ~S = 1

µ0

~E × ~B

Math Review

sin θ = (opp)/(hyp), cos θ = (adj)/(hyp) tan θ = (opp)/(adj)

Pythagorean theorem: (hyp)2 = (opp)2 + (adj)2

Sphere: V =4

3πr3, A = 4πr2

Vectors(A or ~A) and components (Ax,Ay,...) ~A· ~B = AB cos θ = AxBx+AyBy

− ~A points opposite ~A ~A − ~B means ~A + (− ~B) | ~A × ~B| = AB sin θComponent = (sign)(magnitude)(trig fn)~C = ~A + ~B means Cx = Ax + Bx and Cy = Ay + By

Magnitude of ~C =∣

∣

∣

~C∣

∣

∣=

√

C2x + C2

y

Angle to nearest x-axis: θ = Tan−1|Cy||Cx|

Specify axis!