If $\vec{\mathbf B}=B\vec{\mathbf a}_z$, compute the magnetic flux passing through a hemisphere of radius $R$ centered at the origin and bounded by the plane $z=0$.
Solution
The hemisphere and the circular disc of radius $R$ form a closed surface, as illustrated in the figure; therefore, the flux passing through the hemisphere must be exactly equal to the flux passing through the disc.
The flux passing through the disc is
$$\Phi=\int_S\vec{\mathbf B}\cdot\mathrm d\vec{\mathbf s}=
\int\limits_0^R\int\limits_0^{2\pi}B\rho\,\mathrm d\rho\,\mathrm d\phi
=\pi R^2B$$
The reader is encouraged to verify this result by integrating over the surface of the hemisphere.
According to Maxwell's equations the magnetic flux over a closed surface must be zero, why in this case does not happen?
Best Answer
The flux through the closed hemisphere is zero, $$\Phi_{\mathrm{hemi}}+\Phi_{\mathrm{disk}} = 0.$$ This allows us to find the flux through the hemisphere knowing the (more easily calculable) flux through the disk, $$\Phi_{\mathrm{hemi}} = -\Phi_{\mathrm{disk}}.$$