Maxwell's Equations — Question 5

Question 5

Faraday feedback requires a radial correction to a quadratic electric ramp

In a source-free local axisymmetric vacuum model, take $B_\phi=\alpha tr/c^2$. A proposed axial electric field is $E_z=\alpha t^2+f(r)$, with $f(0)=0$. There are no other components.

1. Use Faraday law to determine f(r).

2. Check the Ampere-Maxwell law and both divergence laws for the resulting fields.

Question 5 - Solution

Faraday feedback requires a radial correction to a quadratic electric ramp

(a) Method and calculation

$$(\nabla\times\mathbf E)_\phi=-\partial_rE_z=-\partial_tB_\phi=-\alpha r/c^2.$$$$\boxed{f(r)=\alpha r^2/(2c^2),\quad E_z=\alpha[t^2+r^2/(2c^2)]}.$$

(b) Interpretation and further calculation

$$(\nabla\times\mathbf B)_z=r^{-1}\partial_r(rB_\phi)=2\alpha t/c^2=c^{-2}\partial_tE_z.$$$$\boxed{\nabla\cdot\mathbf E=0,\quad\nabla\cdot\mathbf B=0}.$$

Physical check

A spatially uniform quadratic ramp alone would satisfy one curl equation but fail the other; the radial correction makes the local model consistent.