Flux of an Electric Field — Question 3

Question 3

A transverse uniform field crosses a cylinder without net side flux

A closed cylinder has radius R, length L and axis z. A uniform electric field is $\mathbf E=E_\perp\hat{\mathbf x}+E_\parallel\hat{\mathbf z}$. Orient every surface outward.

1. Find the flux through each end and through the curved side.

2. For $E_\perp>0$, integrate the positive outward flux over the half of the curved side where $\cos\phi>0$. Compare it with the net side flux.

Question 3 - Solution

A transverse uniform field crosses a cylinder without net side flux

(a) Method and calculation

$$\Phi_{\mathrm{top}}=E_\parallel\pi R^2,\quad\Phi_{\mathrm{bottom}}=-E_\parallel\pi R^2.$$$$\Phi_{\mathrm{side}}=E_\perp RL\int_0^{2\pi}\cos\phi\,d\phi=0.$$

(b) Interpretation and further calculation

$$\Phi_{\mathrm{side,+}}=E_\perp RL\int_{-\pi/2}^{\pi/2}\cos\phi\,d\phi=2E_\perp RL.$$$$\Phi_{\mathrm{side,-}}=-2E_\perp RL,\quad\Phi_{\mathrm{closed}}=0.$$A zero side integral arises from cancellation, not from a field tangent everywhere.

Physical check

The transverse field is normal to the cylinder at some points and tangent at others.