Gauss' Law for Magnetic Fields — Question 3

Question 3

A hemisphere flux can be computed from its closing disk

A hemispherical dome of radius a lies above the xy-plane in uniform field $\mathbf B=B\hat z$. The dome normal points outward from the enclosed half-ball.

1. Find flux through the curved dome without integrating over its area.

2. Compare mean normal field on the dome with B.

Question 3 - Solution

A hemisphere flux can be computed from its closing disk

(a) Method and calculation

$$\Phi_{\mathrm{disk}}=-B\pi a^2,\quad\Phi_{\mathrm{dome}}+\Phi_{\mathrm{disk}}=0.$$$$\boxed{\Phi_{\mathrm{dome}}=B\pi a^2}.$$

(b) Interpretation and further calculation

$$A_{\mathrm{dome}}=2\pi a^2\Rightarrow\langle B_n\rangle=\Phi_{\mathrm{dome}}/A_{\mathrm{dome}}.$$$$\boxed{\langle B_n\rangle=B/2}.$$

Physical check

Using curved surface area times B would ignore the varying normal direction.