EM equations + more Flashcards

1
Q

What are Maxwell’s equations?

A

-

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2
Q

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

A

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

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

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3
Q

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

A

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

B is curl of A

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4
Q

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

A

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

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5
Q

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

A

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

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6
Q

What is the lorentz force law?

A

force = charge * (E + v cross B)

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7
Q

What is the continuity equation for charge?

A

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

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8
Q

What is the equation for the Poynting vector?

A

1/mu0 * (E cross B)

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9
Q

What is the equation for EM energy density?

A

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

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10
Q

What are the electric and magnetic potential dipole terms?

A

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)

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11
Q

What is the 4-differential?

A

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

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

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12
Q

What are the terms in the 4-momentum?

A

contravariant 4-momentum =

(E/c , 3-momentum)

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

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13
Q

What is the magnitude of the 4-momentum?

A

m^2 c^2

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14
Q

What is the relativistic kinetic energy of a particle?

A

(gamma - 1) m c^2

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15
Q

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

A

E = gamma m c^2

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

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16
Q

What are the components of the 4-current?

A

charge density * 4-velocity

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

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17
Q

What are the components of the 4-potential?

A

contravariant: (V/c , A)

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18
Q

What is the 4-wave operator?

What is the 4-wave equation?

A

4-wave operator: contravariant and covariant 4-derivatives

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

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19
Q

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

A

-

20
Q

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

A

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.

21
Q

What form does the retarded time take?

A

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

22
Q

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

A

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 ]

23
Q

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

A

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 ]

24
Q

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

A

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.

25
Q

What is the uniqueness theorem for Laplace’s equation?

What are sufficient boundary conditions?

A
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.

26
Q

What is the divergence theorem?

A

Integral of the divergence of a vector field over a volume

The flux of that field through the enclosing surface.

27
Q

What is Stoke’s theorem?

A

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.

28
Q

What is the coulomb gauge condition?

A

Div(A) = 0

29
Q

How can we calculate the electric dipole moment?

How about the magnetic dipole moment?

A

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

30
Q

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

A

Energy approximately equal to pc

Kinetic energy approximately equal to gamma m c^2

31
Q

What is the volume of a sphere?

What about surface area?

A

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

32
Q

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

A

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

33
Q

What is the integral expression for the electric potential?

A

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)

34
Q

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

A

v^2 / r

35
Q

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

A

some constant * cos(kx - wt)

w = kc

36
Q

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

A

qV

37
Q

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

A

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

38
Q

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

A

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

39
Q

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

A

mu down, nu along

40
Q

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

A

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

41
Q

What is the gauge invariance in electrodynamics?

A

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

A grad of a scalar function can be added to A.

42
Q

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

A

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.

43
Q

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

A

Time measured in the frame = gamma* the proper time

44
Q

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

A

-

45
Q

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

A

-