Gauss' Law for Magnetic Fields — Question 10

Question 10

An axial field that weakens along a tube requires radial spreading

Near the axis of a cylindrically symmetric field, assume $B_z=B_0(z)$ is independent of r and there is no azimuthal component. The radial component is finite at r=0. Use $\nabla\cdot\mathbf B=r^{-1}\partial_r(rB_r)+\partial_zB_z$.

1. Find Br in terms of the axial gradient.

2. For $B_0(z)=B_*e^{-z/\ell}$, find Br and its direction for positive z and positive Bstar.

Question 10 - Solution

An axial field that weakens along a tube requires radial spreading

(a) Method and calculation

$$\partial_r(rB_r)=-rB_0^{\prime}(z)\Rightarrow rB_r=-r^2B_0^{\prime}/2+f(z).$$$$\boxed{B_r=-\tfrac r2B_0^{\prime}(z)}\quad\text{since finiteness requires }f=0.$$

(b) Interpretation and further calculation

$$B_0^{\prime}=-B_0/\ell.$$$$\boxed{B_r=\frac{rB_0(z)}{2\ell}>0\quad\text{(outward)}}.$$

Physical check

Axial weakening is accompanied by transverse flux spreading, preserving zero magnetic divergence.