Question

Starting with Maxwell's equations show that the magnetic field satisfies the same wave equation as the electric field. in particular, that is, too, propagates with the same speed.

Answer 1

The Maxwell equations can be written as:

where is the density per area unit, is the conductivity of the media, is the magnetic permeability is the electrical permitivity and   is the gradient operator defined by

The ecuation (1) is the Gauss Law, (2) the magnetic Gauss Law, (3) Faraday's Law and (4) Ampere- Maxwell Law.

let's apply to equation (4) the rotational operator like

replacing the rotational terms in the equation with the Faraday Law eq (3) we have

Solving the left part of the equation using the cross product identities we get

The first term of this identity is zero. This leads to

The same analisys can be made fro equation (3), applying the rotational operator to the Faraday's Law

Replacing the Ampere-Maxwell law we get

Solving the left part of the equation we have

replacing in (6) we have

If we consider that we are in vaccum that imply that and we don't have any current that gives . With this two conditions the equations (5) and (7) take the form

This satifices for both the E and B field the wave equations, where the velocity of the propagation is

Where c is the speed of light in the vaccum