Lorentz transformation of the electromagnetic fields

Problem:

A long superconducting solenoid with radius R is at rest in frame K.  It has its axis along the z-axis and a current flowing on the surface produces a uniform field inside.
(B = Bk, r < R; B = 0, r > R.)

image
An observer moves with uniform velocity v = vi (v << c) along the x-axis.  Write down the electric and magnetic fields in the rest frame K' of the observer for r < R and r > R.

Solution:

Problem:

A neutral conducting sphere is at rest with its center at the origin in reference frame K.  A uniform magnetic field B = B0k is present.  Reference frame K' moves with uniform velocity v = vi with respect to K.  Find E an B inside the sphere as observed in K'.

Solution:

Problem:

The infinite xy plane is a non-conducting surface with surface charge density σ, as measured by an observer at rest on the surface.  A second observer moves with velocity vi in the positive x-direction with respect to the surface at height h above it.  Find an expression for the electric field and magnetic field measured by this observer.

Solution:

Problem:

A proton of mass m0 moving with speed βz in the z-direction, encounters a quadrupole magnetic field of the form B = B0xj + B0yi over a length L along the z-direction.  Both the magnetic field B and the length L are as observed in the laboratory frame.
(a)   Find the electric field experienced by the particle as a function of x and y in its rest frame.
(b)   If the magnet is short enough, such that the acceleration of the particle going through the field can be considered an impulse, (i.e. the acceleration changes the direction, but not the position), find the impulse observed in the lab and the entering particle's rest frame, for a particle that enters the magnet at x = x0, y = 0.  Explain the difference.

Solution:

Problem:

Plot E and B as a function of time at a point P one cm away from the path of a 10 MeV proton.  Set P at (0, 0.01, 0) with the charge at (-vt, 0, 0).

Solution:

Problem:

A 30 GeV proton passes 10-7cm away from a hydrogen atom.
(a)  Estimate the peak magnitude of the electric field and the duration of the electric field pulse to which the atom is subjected.
(b)  Do the same for a 30 GeV electron passing at the same distance.
You may use mpc2 = 1 GeV and mec2 = 0.5 MeV.

Solution:

Problem:

A fixed dipole moment is pointing in the x-direction while moving in the z-direction with a constant velocity v << c.  What are the instantaneous electric and magnetic fields at a point (x, y = 0, z) away from the dipole?

image

 

Solution:

Problem:

A line of charge with charge density λ C/m is fixed at rest along the x' axis of a reference frame S'.  A test charge q is at rest in S' at (0, 0, z' = d).  S' is in constant motion with velocity v = vi with respect to a reference frame S.
(a)  Calculate the electric field of the line of charge in the rest frame S' and the force on q. 
(b)  Calculate the electric and magnetic fields of the line of charge measured by an observer at rest in S.
(c)  Calculate the force measured by the observer in S on the test charge q.

Solution: