Date fre 26 januari 2018

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

$$ \phi(r,t) = \int_S exp(i w |r|\theta + i w t) Y_{l,m} dS $$

We could design a vector potential as

$$ A(K(|r|,e),r,t) = (\int_S K(|r|,e) exp(i w r \cdot e + i w t) Y_{l,m} dS $$

Note that we have

$$ \frac{\partial}{\partial x_i} A(K,r,T) = A(i w e_i K(|r|,e),r,t) $$

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

$$ H(K,r,t) = C_1 \nabla \times A(K,r,t) = C_1 w i A(e \times (aK_1+bK_2),r,t) $$

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

$$ \nabla \times \nabla H - w^2r_0^2 H = 0 $$

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

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

$$ J = C_3 H $$

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

$$ J = C_4 A(e\times\hat z,r,e) = C_5 \nabla\times\hat z j_l(W |r|) Y_{lm}exp(iwt) $$

which becomes

$$ J = C_6 j_l'(wr_0)r\times \hat z Y_{lm} $$

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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