Electromagnetism

what is the equation that the dirac delta obeys

integral of d³x f(t,x)delta³(x-z) = f(t,z)

what is the continuity equation

Dtp(t,x) + triange . t(t,x) = 0

what does rearranging the continuity equation tell us

the rate of change of charge in a volume is due entirely to current flowing in/out of that volume through its surface

what is charge

a point particle with electric charge

what is current

a moving charge

what is p(t,x) (wonky p)

charge density

what is charge density

amount of charge per unit volume

what is J

current density

what is equation for charge density of moving charge on trajectory

p(t,x) = qdelta³(x - T)

what is the equation for current density of a moving charge on a trajectory

J = qT(t)delta³(x- T(t))

what is the integral form of the continuity equation

DtQinsidev = -integral DV DS.J

what is Gausss law

triangle. E(t,x) = 1/E0 * p(t,x)

what is Faradays Law of inductance

triange x E(t,x) = -DB(t,x)/Dt

what is Gauss’s law for magnetism

triangle . B(t,x) = 0

what is Amperes law

triangle x B(t,x) = u0J(t,x) + u0E0DE(t,x)/Dt

what is the lorentz force equation

mx’’(t) = qE(t,x(t)) + qx’(t) x B(t,x(t))

what are the equations of electrostatics

triangle. E(x) = p(x)/E0 and triangle x E(x) = 0

what is the equation for the scalar field

it is - triangle²scalar(x) = p(x)/E0

what are the fourier conventions

f(k) = integral d³xe^ikx f(x) and f(x) = integral d³k/(2pi)³ e^-ikx f(k)

what is the greens function solution to scalar field and electric field

scalar(x) = 1/4piE0 integral d³y p(y)//x-y/ and E(x) = 1/4piE0 integral d³y x-y//x-y/³ p(y)

what is the equation for scalar field and electric fied for p(y) = qdelta³(y-z)

scalar(x) = q/4piE0x-z/ and E(x) = q/4piE0 (/x-z/ / /x-z/³)

what is Gauss law in integral form

Q = inetrgal V of d³xp(x) = E0 integral DS.E

what is the electric field outside any spherically symmetric charge distribution

E(x) = Q/4piE0/x/² x^

what is the electrostatic energy stored in an electric field

U = E0/2 integral d³x E(x) E(x)

what is the taylor series of a multivariative function f about a point x

f(x+y) = sum from n=0 to infinity of 1/n! (y.trianglex)^nf(x) = f(x) + y.trianglexf(x) + 1/2! y^iy^jDiDjf(x)

what is the potential of an electric dipole

potential(x) = 1/4piE0 p.x/ /x/³ where p is the dipole moment

what is the force for a point charge

F = qE

what is the force for a dipole

F= p. triangle(E)

what is the electrostatic energy of a charge q at x

U(x) = qscalar(x)

what is the electrostatic energy of a dipole at x

U(x) = p. triangle scalar(x)

what is the general force equation

F(x) = -triangle U(x)

what is the multipole expansion equation

scalar(x) = Q0/4piE0/x/ + x.p/4piE0/x/³ + ½ x^TQx/4piE0/x/^5 + ….

what is Q0

Q0 = integral of d³y p(y)

when is the charge Q0

if we are sufficiently far from the charge distribution

what is the meaning of p in the second term of the multipole expansion

p = integral of d³y yp(y)

what does the second term of the multipole expansion mean

probes the spatial structure of the charge distribution

what is the equation for the quadrupole moment

Qij = integral of d³y(3y^iy^j - delta^ijy^ky^k)p(y)

what is the distribution of a quadrupole

either two dipoles or 4 charges

what is the equation for the discontinuity of the electric field across the surface charge

n.E(x+) - n.E(x-) = surface charge denisty/ E0

what is the equation for the continuity of the electric field across the surface charge

n x E(x+) - n x E(x-) = 0

BOUNDARY CONDIITONS?

i dinne know

what is the electric field and potential for a conductor

inside a conductor the electric field is 0 and the scalar potential is constant

what are the equations of magnetostatics

triangle x B(x) = u0J(x) and triangle.B(x) = 0

what is triangle.J in magnetostatics

0 because of continuity equation since p’ = 0 and required in faradays law because triangle. (triangle x) anything is 0

what is the gauge transformation

A(x) and A’(x) = A(x) - trianglex(x)

what is the equation of magnetostatics in terms of gauge of transformation

it is -triangle²A(x) = u0J

what is the biot savart law

A(x) = u0I/4pi integral c d³y J(y)/ /x-y/ and B(x) = u0I/4pi integral c d³y J(y) x (x-y)/ /x-y/³

what is the biot savart law for a current in an infinitesimally thin wire

A(x) = u0I/4pi integral c dy/ /x-y/ and B(x) = u0I/4pi integral c dy x (x-y)/ /x-y/³

what is the integral form of Amperes law

integral closed c of dx.B = u0I

what is the magnetostatic energy stored in a magnetic field

U = 1/2u0 integral d³x B(x).B(x)

what is the vector potential of a magnetic dipole

A(x) = u0/4pi m x x/ /x/³ where m is the magnetic dipole moment

what is the equation for the magnetic dipole moment in multipole expansion

m = ½ integral d³y y x J(y)

what is the magnetic field corresponding to the vector potential of a magnetic dipole

B(x) = u0/4pi (3(m.x^)x^ - m / /x/³)

what is the equation for the discontinuity crossing surface of magnetic field

n x B(x+) - n x B(x-) = u0K(x)

what is the equation for the continuity crossing a magnetic field

n.B(x+) - n.B(x-) = 0

how is magnetic field and vector potential related

B = triangle x A

what is triangle. B

0 so there is no magnetic charge

what is the flux through the loop bounding a surface S

flux = integral s DS.B

what is the emf

emf = integral C dx( E + v x B) where v is the velocity of the loop

how is emf and flux related

emf = - Dt flux

what is lenzs law

an induced EMF opposes the change producing it, nature abhors a change of flux, there is a minus sign in the law of inductance

what is the equation for self inductance

L = flux/ I(t) where flux is due to the magnetic field of the loop

what is the equation for back EMF

EMF = - Dt flux = -L Dt I

what is ohms law

for current inside conducting materials J = @E where @ is conductivity

how does self induction work

faradays law states time dependent magnetic field induces electric field, so if a conducting wire is placed in changing B then current is driven, the current moves through a loop of wire generating time independent magnetic field

what is the equation for electric field in gauge transformation

E = - triangle(scalar potential) - Dt A

what is a general gauge transformation

A to A - triangle V and scalar potential to scalar potential + V’

what can we choose V to be

E0u0 sclar potential’ + triangle A = 0

what do the maxwell equations reduce to in the gauge transformations

1/c² Dt² scalar potential - triangle² scalar potential = p/E0 and 1/c²Dt²A - triangle²A = u0J

what is the general solution of the wave equation

A(t,x) = integral d³k/(2pi)³(2w) [A(k) e^-iwt-ikx + A*(k)e^iwt+ikx], scalar potenital is same replacing A and A* with scalar potential where w = c/k/

when do electric and magnetic fields obey the wave equation

in a vacuum