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!
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