EM equations + more

Question 1

What are Maxwell’s equations?

Answer 1

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Question 2

What is the lorentz transformation of acceleration in the bremsstrahlung and synchrotron schema?

Answer 2

bremsstrahlung: gamma^3 factor
synchrotron: gamma^2 factor

(which is rest frame? (i think primed frame))

Question 3

What are the equations for the electric and magnetic fields in terms of their potentials?

Answer 3

E is -(grad V) - time derivative of A

B is curl of A

Question 4

What is the lorentz gauge condition for the electric and magnetic potentials?

Answer 4

1/c^2 (time derivative of V) + div(A) = 0

Question 5

What are the wave equations for the electric and magnetic potentials?

Answer 5

wave operator on V gives rho/ epsilon0
wave operator on A gives mu0 * current density

wave operator= 1/c^2 second time derivative - laplacian operator

Question 6

What is the lorentz force law?

Answer 6

force = charge * (E + v cross B)

Question 7

What is the continuity equation for charge?

Answer 7

Divergence of current density + time derivative of charge density = 0

Question 8

What is the equation for the Poynting vector?

Answer 8

1/mu0 * (E cross B)

Question 9

What is the equation for EM energy density?

Answer 9

u = (e0 / 2) lEl^2 + (1/2*mu0) lBl^2

Question 10

What are the electric and magnetic potential dipole terms?

Answer 10

electric: 1/(4 pi e0 r^2) * (electric dipole moment) dot (r-vector)
magnetic: mu0/(4 pi r^2) * (magnetic dipole moment) cross (r-vector)

Question 11

What is the 4-differential?

Answer 11

Remember contravariant differential is differentiating with respect to the covariant 4-position and vice-versa.

contravariant 4-differential = (1/c time derivative, del)

Question 12

What are the terms in the 4-momentum?

Answer 12

contravariant 4-momentum =

(E/c , 3-momentum)

= (gamma m c, gamma m 3-velocity)

Question 13

What is the magnitude of the 4-momentum?

Answer 13

m^2 c^2

Question 14

What is the relativistic kinetic energy of a particle?

Answer 14

(gamma - 1) m c^2

Question 15

What are two forms of the relativistic energy of a particle?

Answer 15

E = gamma m c^2

E^2 = (pc)^2 + (mc^2)^2

Question 16

What are the components of the 4-current?

Answer 16

charge density * 4-velocity

{4-velocity = (gamma * c, gamma * 3-velocity)}

Question 17

What are the components of the 4-potential?

Answer 17

contravariant: (V/c , A)

Question 18

What is the 4-wave operator?

What is the 4-wave equation?

Answer 18

4-wave operator: contravariant and covariant 4-derivatives

4-wave equation: 4-wave operator acting on the 4-potential = mu0 * the 4-current

Question 19

What are the lorentz transformations of the electric and magnetic fields?

Answer 19

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Question 20

How would you go about showing the angular power distribution for a charge oscillating in space?

Answer 20

Find the time-dependent form of the charge position. Differentiate to get the acceleration and take the mean square value. Use this in the Larmour formula.

Question 21

What form does the retarded time take?

Answer 21

The time experienced by the source minus the (distance from charge at the retarded position)/(speed of light)

Question 22

What is the continuity equation for a magnetic field crossing a boundary?

Answer 22

The magnetic field perpendicular to the surface stays constant.

The magnetic field parallel to the surface is larger outside the surface by an amount:
[ mu0 * the surface current density ]

Question 23

What is the continuity equation for an electric field crossing a boundary?

Answer 23

The electric field parallel to the surface stays constant.

The electric field perpendicular to the surface is larger outside the surface by an amount:
[ the surface charge density / e0 ]

Question 24

What is the condition Laplace’s equation places on the potential which obeys it?

Answer 24

The potential can have no local maxima/minima in regions obeying Laplace’s equation (regions free of charge).
In this region, the potential at r is equal to the average of potential at points equidistant from r.

Question 25

What is the uniqueness theorem for Laplace's equation? | What are sufficient boundary conditions?

Answer 25

``` Given sufficient b.c.'s : -V at the boundary or -flux of V through the boundary {grad(V) dot nhat} (i.e. E perpendicular to the boundary) ``` there is only one solution to Laplace's equation.

Question 26

What is the divergence theorem?

Answer 26

Integral of the divergence of a vector field over a volume = The flux of that field through the enclosing surface.

Question 27

What is Stoke's theorem?

Answer 27

The surface integral of ( The curl of a vector field ) parallel to the surface. = The line integral of the vector field parallel to the line enclosing that surface.

Question 28

What is the coulomb gauge condition?

Answer 28

Div(A) = 0

Question 29

How can we calculate the electric dipole moment? | How about the magnetic dipole moment?

Answer 29

electric: volume integral of {charge density * r-vector } magnetic: equal to the current through a loop * the area vector of the loop

Question 30

What is the is the energy for an Ultra-relativistic particle? What about its kinetic energy?

Answer 30

Energy approximately equal to pc | Kinetic energy approximately equal to gamma m c^2

Question 31

What is the volume of a sphere? | What about surface area?

Answer 31

Volume: 4/3 pi r^3 SA: 4 pi r^2

Question 32

What is the equation representing the local conservation of electromagnetic energy?

Answer 32

Work done on a charge (E dot j) = - (Div Poynting vector) - time derivative of energy density

Question 33

What is the integral expression for the electric potential?

Answer 33

1/ (4 pi e0) * volume integral of charge density/ distance from charge density distance from charge density R = r - r' (vector expression where r is the field point and r' is the source location)

Question 34

What is the acceleration of a point moving in a circular orbit?

Answer 34

v^2 / r

Question 35

What is a choice you could make for the E component of an electromagnetic wave?

Answer 35

some constant * cos(kx - wt) | w = kc

Question 36

What is an expression for the potential energy of a charge q in a potential V?

Answer 36

qV

Question 37

What is the integral form of the maxwell equation involving the cross product of E?

Answer 37

The line integral of the E field = - (the rate of change of magnetic flux through the corresponding surface)

Question 38

What is the integral form of the maxwell equation involving the cross product of B?

Answer 38

The line integral of the B field = mu0 * the enclosed charge + e0 mu0 * (the rate of change of electric flux through the surface)

Question 39

Which way do mu and nu go in the strength tensor?

Answer 39

mu down, nu along

Question 40

How can you compute the lorentz transform of the strength tensor?

Answer 40

Apply the LAMBDA operator twice on the two different indices. Equal to the matrix calculation: LAMBDA * F * LAMBDAtranspose

Question 41

What is the gauge invariance in electrodynamics?

Answer 41

A time derivative of a scalar function can be added to V. A grad of a scalar function can be added to A.

Question 42

What condition do we need on the poynting vector to constitute radiation?

Answer 42

We need the flux integral of the poynting vector (energy flux) to go to a constant, even as the sphere of integration gets infinitely large.

Question 43

What is the relationship between the proper time and the time measured in some frame?

Answer 43

Time measured in the frame = gamma* the proper time

Question 44

Draw a diagram for the radiation emitted in the bremsstrahlung scheme.

Answer 44

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Question 45

Draw a diagram for the radiation emitted in the synchrotron scheme.

Answer 45

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