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.