Electric field

Field parameter

Charge density

div(grad(V))= - charge density/epsilon

V = volume integrate(charge density/distance)

Potential
V = k*volume integrate(charge density/distance)

E=-grad(V)

V=-line integrate (E) from reference to point

Field strength
E =k* volume integrate (charge density / square(distance))

div(E)=charge density/epsilon, curl(E)=0

Boundary condition

difference(normal(E)) = charge area density/epsilon
difference(parallel(E)) = 0

difference(direaction derivative(V))=-charge area density/epsilon

Work and Energy

Work
W=-line integrate(q * E) from state to end
W=difference(q * V)

Potential energy
W = 1/2 * volume integrate(charge density * V)
W = 1/2 * volume integrate(epsilon * square(E))

Energy cannot be superposition

Electrostatic pressure
Conductor has charge, under electric field, it will exerted on force
F=1/2 * charge area density * (Eabove + E below)

Laplace equation

General soluton:

Boundary Conditions and Uniqueness theoerms
The solution to Laplace's equation in some volume
V is uniquely determined if V is specified on the boundary surface S.

Conductors and Second Uniqueness Theorem
In a volume V surrounded by conductors and
containing a specified charge density p, the electric field is uniquely determined if the total charge on each conductor is given. (The region as a whole can be bounded by another conductor, or else unbounded.)

The Method of Images

Assume that everything is the same in the two problems. Energy, however, is not the same.

Separation of Variables

Multipole Expansion
V(r)=k*int(charge density/|r-r'|)
1/|r-r'|=sum(r^-(n+1)*r'^n*Pn(cos(theta)))

multipole

P0(x)=1
P1(x)=x
P2(x)=1/2*(3x^2-1)

monopole

V(r)=k/r int(charge density)

dipole moment

V(r)= k*dot(p,r)/r^2
p = int(r'*charge(r'))

E=k/r^3 [3*dot(p, r)r-p]

quardole moment

V(r)=k*sum(r_i, r_j, Q_ij)/2r^3
Q_ij=int((3*r_i*r_j-r_i*r_j*delta_ij)*charge density)

Single dipole

Torque

N=cross(p,E)

Force
dipole under the nonuniform field E will
be exerted force

F=dot(p,grad(E))

Potential Energy

U=-dot(p, E)

Potential energy for two dipoles
U=k/r^3 [3*dot(p1, r)dot(p2, r)-dot(p1,p2)]

Net dipole moment
P=sum(p_i)

Potential

V=k*int(dot(r, P(r'))/r^2)

bounded charge density

-div(P)

bounded charge area density

dot(P, n)