Induced Magnetic Fields — Question 10

Question 10

Local curl distinguishes a changing-field region from its exterior

An axisymmetric vacuum field has $B_\phi=kr$ for r<a and $B_\phi=ka^2/r$ for r>a. No conduction current is present. Use $(\nabla\times\mathbf B)_z=r^{-1}\partial_r(rB_\phi)=c^{-2}\partial_tE_z$.

1. Find the electric-field rate in both regions.

2. Check continuity of B at a and explain why nonzero exterior B does not imply nonzero local electric-field rate.

Question 10 - Solution

Local curl distinguishes a changing-field region from its exterior

(a) Method and calculation

$$r<a:\quad r^{-1}\partial_r(kr^2)=2k\Rightarrow\boxed{\partial_tE_z=2kc^2}.$$$$r>a:\quad r^{-1}\partial_r(ka^2)=0\Rightarrow\boxed{\partial_tE_z=0}.$$

(b) Interpretation and further calculation

$$B_\phi(a^-)=ka=B_\phi(a^+).$$$$\boxed{B_\phi(r>a)\ne0\quad\text{but}\quad\nabla\times\mathbf B=0\ \text{there}}.$$

Physical check

The exterior circulation encloses an interior source of curl, just as a current-free exterior can surround a wire.