Why the magnetic flux is not zero?

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?

• It would be great if someone could transcribe the text from the image, and crop the image to just the figure itself. May 20, 2013 at 5:19
• @DavidZaslavsky is there some way to get the text next to the image (rather than underneath it) Jul 8, 2013 at 4:07
• @UnkleRhaukus no, not on this site. That's okay though, the exact formatting isn't important for these things. Jul 8, 2013 at 5:43
• You may have misunderstood the question in the manner of physics.stackexchange.com/questions/24609/… Nov 8, 2014 at 1:15

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, $$\Phi_{\mathrm{hemi}} = -\Phi_{\mathrm{disk}}.$$