Induced Electric Fields — Question 7

Question 7

An added electrostatic field can cancel induction locally without changing its circulation

Inside a cylinder, a uniform axial field increases at rate beta. Its induced electric field is $\vec E_{\mathrm{ind}}=(\beta/2)(y\hat x-x\hat y)$. Add a uniform electrostatic field $E_0\hat x$, with E0 and beta positive.

1. Find the point where the total electric field vanishes, assuming it lies inside the cylinder of radius a.

2. Find the circulation around a concentric circle of radius r<a and the condition for that zero-field point to lie inside.

Question 7 - Solution

An added electrostatic field can cancel induction locally without changing its circulation

(a) Method and calculation

$$E_x=E_0+\beta y/2=0,\quad E_y=-\beta x/2=0.$$$$\boxed{(x,y)=(0,-2E_0/\beta)}.$$

(b) Interpretation and further calculation

$$\oint\vec E\cdot d\vec\ell=\oint\vec E_{\mathrm{ind}}\cdot d\vec\ell+E_0\oint dx=\boxed{-\pi r^2\beta}.$$$$\boxed{2E_0/\beta<a}.$$

Physical check

Zero total field at one point does not remove a nonzero circulation around a larger contour.