Motion in the presence of electric and magnetic fields
The Poynting vector
S = (1/μ0)(E × B) is the energy flux.
∇·E = ρ/ε0, ρ = ε0∂Ez/∂z = ε0k.
Motional emf: I = emf/R
emf = vb(B1 - B2) = vb[μ0I/(2π)] (1/r - 1/(r+a))
Thevenin equivalent circuit
Any two terminal network containing only voltage sources,
current sources, and other resistors can be replaced by a voltage source
Vth in series with a resistor Rth.
To find Vth, calculate the output voltage VAB for an open
circuit between A an B.
To find Rth, replace voltage sources with short circuits and current sources with open circuits. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, "looking back" into the circuit. This is Rth. Here Rth = R1R2/(R1 + R2),
The time constant of the circuit is t = RthC.
Method of images, vector addition
Ep = [2q/(4πε0R2)][d/R]
Ohm's law, ΔV = IR
ρ = ε0∇·E ∝ ∇·(r/r5/2)
= (1/r2)∂(r2*r-3/2)/∂r ∝ ∇·r-/2.
Or use Gauss' law in integral form.
4πr2E(r) = Qinside/ε0. Qinside ∝ r1/2, dQinside/dr ∝ r-1/2.
dQinside/dr = 4πr2ρ(r)dr, ρ(r) ∝ r-5/2.
The divergence of a curl is zero.
Boundary conditions for electromagnetic fields
D1)·n2 = σf,
(B2 - B1)·n2 = 0,
(E2 - E1)·t = 0,
(H2 - H1)·t = kf∙n.