Maxwell's Equations — Question 6

Question 6

A prescribed nonuniform magnetic ramp determines an electric curl and a current

In a region let $B_z=\beta xt$, $B_x=B_y=0$, and seek $\mathbf E=E_y(x)\hat y$ with $E_y(0)=0$. Use vacuum Maxwell equations.

1. Find Ey from Faraday law.

2. Find rho and J needed for this field pair.

Question 6 - Solution

A prescribed nonuniform magnetic ramp determines an electric curl and a current

(a) Method and calculation

$$\partial_xE_y=-\partial_tB_z=-\beta x\Rightarrow\boxed{E_y=-\beta x^2/2}.$$$$\nabla\cdot\mathbf E=\partial_yE_y=0\Rightarrow\boxed{\rho=0}.$$

(b) Interpretation and further calculation

$$(\nabla\times\mathbf B)_y=-\partial_xB_z=-\beta t,\quad\partial_t\mathbf E=0.$$$$\boxed{\mathbf J=-(\beta t/\mu_0)\hat y,\quad\nabla\cdot\mathbf J=0}.$$

Physical check

A changing B need not require a changing E; a static electric curl can accompany a constant magnetic ramp rate.