EM equations + more Flashcards
What are Maxwell’s equations?
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What is the lorentz transformation of acceleration in the bremsstrahlung and synchrotron schema?
bremsstrahlung: gamma^3 factor
synchrotron: gamma^2 factor
(which is rest frame? (i think primed frame))
What are the equations for the electric and magnetic fields in terms of their potentials?
E is -(grad V) - time derivative of A
B is curl of A
What is the lorentz gauge condition for the electric and magnetic potentials?
1/c^2 (time derivative of V) + div(A) = 0
What are the wave equations for the electric and magnetic potentials?
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
What is the lorentz force law?
force = charge * (E + v cross B)
What is the continuity equation for charge?
Divergence of current density + time derivative of charge density = 0
What is the equation for the Poynting vector?
1/mu0 * (E cross B)
What is the equation for EM energy density?
u = (e0 / 2) lEl^2 + (1/2*mu0) lBl^2
What are the electric and magnetic potential dipole terms?
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)
What is the 4-differential?
Remember contravariant differential is differentiating with respect to the covariant 4-position and vice-versa.
contravariant 4-differential = (1/c time derivative, del)
What are the terms in the 4-momentum?
contravariant 4-momentum =
(E/c , 3-momentum)
= (gamma m c, gamma m 3-velocity)
What is the magnitude of the 4-momentum?
m^2 c^2
What is the relativistic kinetic energy of a particle?
(gamma - 1) m c^2
What are two forms of the relativistic energy of a particle?
E = gamma m c^2
E^2 = (pc)^2 + (mc^2)^2
What are the components of the 4-current?
charge density * 4-velocity
{4-velocity = (gamma * c, gamma * 3-velocity)}
What are the components of the 4-potential?
contravariant: (V/c , A)
What is the 4-wave operator?
What is the 4-wave equation?
4-wave operator: contravariant and covariant 4-derivatives
4-wave equation: 4-wave operator acting on the 4-potential = mu0 * the 4-current
What are the lorentz transformations of the electric and magnetic fields?
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How would you go about showing the angular power distribution for a charge oscillating in space?
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.
What form does the retarded time take?
The time experienced by the source minus the (distance from charge at the retarded position)/(speed of light)
What is the continuity equation for a magnetic field crossing a boundary?
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 ]
What is the continuity equation for an electric field crossing a boundary?
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 ]
What is the condition Laplace’s equation places on the potential which obeys it?
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.
What is the uniqueness theorem for Laplace’s equation?
What are sufficient boundary conditions?
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.
What is the divergence theorem?
Integral of the divergence of a vector field over a volume
The flux of that field through the enclosing surface.
What is Stoke’s theorem?
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.
What is the coulomb gauge condition?
Div(A) = 0
How can we calculate the electric dipole moment?
How about the magnetic dipole moment?
electric: volume integral of {charge density * r-vector }
magnetic: equal to the current through a loop * the area vector of the loop
What is the is the energy for an Ultra-relativistic particle?
What about its kinetic energy?
Energy approximately equal to pc
Kinetic energy approximately equal to gamma m c^2
What is the volume of a sphere?
What about surface area?
Volume: 4/3 pi r^3
SA: 4 pi r^2
What is the equation representing the local conservation of electromagnetic energy?
Work done on a charge (E dot j) = - (Div Poynting vector) - time derivative of energy density
What is the integral expression for the electric potential?
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)
What is the acceleration of a point moving in a circular orbit?
v^2 / r
What is a choice you could make for the E component of an electromagnetic wave?
some constant * cos(kx - wt)
w = kc
What is an expression for the potential energy of a charge q in a potential V?
qV
What is the integral form of the maxwell equation involving the cross product of E?
The line integral of the E field = - (the rate of change of magnetic flux through the corresponding surface)
What is the integral form of the maxwell equation involving the cross product of B?
The line integral of the B field = mu0 * the enclosed charge + e0 mu0 * (the rate of change of electric flux through the surface)
Which way do mu and nu go in the strength tensor?
mu down, nu along
How can you compute the lorentz transform of the strength tensor?
Apply the LAMBDA operator twice on the two different indices.
Equal to the matrix calculation:
LAMBDA * F * LAMBDAtranspose
What is the gauge invariance in electrodynamics?
A time derivative of a scalar function can be added to V.
A grad of a scalar function can be added to A.
What condition do we need on the poynting vector to constitute radiation?
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.
What is the relationship between the proper time and the time measured in some frame?
Time measured in the frame = gamma* the proper time
Draw a diagram for the radiation emitted in the bremsstrahlung scheme.
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Draw a diagram for the radiation emitted in the synchrotron scheme.
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