Consider the uniform wave packet, \(c=1\) and plane wave packets as:

We could design a vector potential as

Note that we have

So if we take \(K_1 = \hat z - e(e\cdot \hat z)\) and \(K_2 = e\times \hat z\) we note that \(K_1,K_2\) both are orthogonal to \(e\) and hence the system with \(K=aK_1+bK_2 + e\) will satisfy Maxwell's equation with the Lorenz gauge. We now can calulate

Noting that \(e \times K_1 = K_2\) and \(e \times K_2 = -K_1\),i*i=-1 we conclude that

That's right, \(H\) is a force free field.

The implication of this is that if we consider the spherical shell a plasme then

Take \(a=0\) then we have for \(|r| = r_0\) with vanishing \(j_l(wr_0)\) e.g. the dericative of \(Y_{lm}\) kan be skiped and he

which becomes

We see that direction is in \(\hat \phi\) and the magnitude is \(sin(\theta)\) which translates to the density in the spherical coordinates so in all we get a vector field of concentric rings \(r\times \hat z = const\) and magnitude determined by \(Y{lm}exp(iwt)\) the solutions is currents in the rings that is just a transport with velocity the speed of light and below.

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