Gauss' Law for Magnetic Fields — Question 2

Question 2

A tilted uniform field has cancelling fluxes through a closed cylinder

A cylinder of cap area A is in uniform field B making angle theta with its axis. Both end caps and the curved side form the closed surface.

1. Find the two cap fluxes using outward normals.

2. Find the net curved-side flux, even though the field crosses parts of the side.

Question 2 - Solution

A tilted uniform field has cancelling fluxes through a closed cylinder

(a) Method and calculation

$$\Phi_{\mathrm{top}}=BA\cos\theta,\quad\Phi_{\mathrm{bottom}}=-BA\cos\theta.$$$$\boxed{\Phi_{\mathrm{top}}+\Phi_{\mathrm{bottom}}=0}.$$

(b) Interpretation and further calculation

$$\Phi_{\mathrm{side}}=-(\Phi_{\mathrm{top}}+\Phi_{\mathrm{bottom}}).$$$$\boxed{\Phi_{\mathrm{side}}=0\quad\text{as a signed integral}}.$$

Physical check

Local entry and exit on the curved surface cancel; the local normal field need not vanish.