Motion in the presence of electric and magnetic fields

https://en.wikipedia.org/wiki/Lorentz_force

The Poynting vector

**S** = (1/μ_{0})(**E** × **B**) is the energy flux.

Maxwell's equations

**∇**·**E** = ρ/ε_{0}, ρ
= ε_{0}∂E_{z}/∂z = ε_{0}k.

Motional emf: I = emf/R

emf = vb(B_{1} - B_{2}) = vb[μ_{0}I/(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
V_{th} in series with a resistor R_{th}.
To find V_{th}, calculate the output voltage V_{AB} for an open
circuit between A an B.

To find R_{th}, 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 R_{th}. Here R_{th} = R_{1}R_{2}/(R_{1}
+ R_{2}),

The time constant of the circuit is t = R_{th}C.

Method of images, vector addition

E_{p} = [2q/(4πε_{0}R^{2})][d/R]

Ohm's law, ΔV = IR

Maxwell's equations

ρ = ε_{0}**∇**·**E** ∝ **∇**·(**r**/r^{5/2})
= (1/r^{2})∂(r^{2}*r^{-3/2})/∂r ∝ **∇**·r^{-/2}.

Or use Gauss' law in integral form.

4πr^{2}E(r)
= Q_{inside}/ε_{0}.
Q_{inside }∝ r^{1/2},
dQ_{inside}/dr_{
}∝ r^{-1/2}.

dQ_{inside}/dr
=
4πr^{2}ρ(r)dr,
ρ(r)
∝ r^{-5/2}.

Vector calculus

The divergence of a curl is zero.

Boundary conditions for electromagnetic fields

(**D**_{2} -
**D**_{1})·**n**_{2} = σ_{f},

(**B**_{2} -
**B**_{1})·**n**_{2} = 0,

(**E**_{2} - **E**_{1})·**t** = 0,

(**H**_{2} - **H**_{1})·**t** =
**k**_{f}**∙n.**