Lorentz transformation of the 4-vector current and potential

Problem:

(a)  State Maxwell's equations in differential form.
(b)  State Maxwell's equations in integral form.
(c)  Carry out the Lorentz transformation for the vector potential

Aμ = (-Ex2/c, -(B/2)x2, (B/2)x1, 0)

along the x1 direction with v = cβ = E/B, (x0, x1, x2, x3) are the usual coordinates.  Note E and B are constants and B ≠ 0.  Find the electric and magnetic fields before and after the Lorentz transformation.

Solution:

Problem:

A point magnetic moment m is at rest in frame K' and in that frame produces a vector potential A' = (μ0/4π)m' x r'/r'3 (SI units) and no scalar potential (Φ = 0).  Frame K' moves with constant velocity v << c along the x-axis of frame K, so that an observer in K sees the moment moving with velocity v = βci.  Show that to first order in β the observer in K detects an electric dipole moment p = β x m/c  as well as an undiminished (to first order) magnetic moment m.

Solution: needs work

Problem:

In reference frame K a long, straight, neutral wire with a circular cross sectional area A = πr2 lies centered on the z-axis and carries a current with uniform current density jk.

image
(a)  Find the Φ, A, E, and B at a point P on the x-axis a distance x > r from the wire.
(b)  In a frame K' moving with velocity vk with respect to K, find the ρ, j, Φ, A, E, and B at the point P.
(c)  In a frame K'' moving with velocity vi with respect to K, find the ρ, j, Φ, A, E, and B at the point P.

Solution:

Problem:

In reference frame K a wire with a circular cross sectional area A = πr2 forms a square of side length L.  The square lies in the xz-plane.  A current with uniform current density j flows clockwise through the wire, I = jπr2.

image

(a)  In a frame K' moving with velocity vi with respect to K, find the charge density ρ' and the current density j'.
(b)  Find the current flowing in the wire in K'.

Solution:

Problem:

Consider a point charge q at rest at the origin in reference frame K.  Reference frame K' moves with velocity v = cβi with respect to K.  At t = 0 the origins of the two frames coincide.
(a) Write down expressions for the 4-vector potential Aμ and the 4-vector current jμ in reference frame K.
(b)  Find A'μ and j'μ in reference frame K'.

Solution: