Constant magnetic field > Magnetic flux > Problem 9.4.5

Statement

$9.4.5.$ A flat horizontal border divides the space into two parts. In the lower part, the magnetic field induction
is zero. Prove that the homogeneous field near the surface in the upper part is directed parallel to it.

Solution

Obviously, the divergence of the field in the lower part of the space is zero, therefore it is also zero at the border, therefore:
$$B_{\perp_{up}} = B_{\perp_{down}} = 0$$
Since there is a field from above, it is exactly parallel to the border.

Answer

Q.E.D.