Constant magnetic field > Magnetic flux > Problem 9.4.3

Statement

$9.4.3.$ Show that the magnetic flux generated by a plane with linear current density $i$ through any closed surface is zero.

Solution

We know that a plane with linear current density $i$ creates a uniform field around itself, its value $$B=\frac{\mu_0 i}{2}$$
, but we don't really need it.

Consider an arbitrary closed surface. The stream is by definition equal to:
$$\oint_S\hat{B}\cdot d\hat{S}$$

The definition of vector divergence will be useful to us.:
$$div \hat{a} = {lim}_{\Delta V\to0} \frac{1}{\Delta V}\oint_S \hat{a}\cdot d\hat{S}=\frac{\partial a}{\partial x}+\frac{\partial a}{\partial y} + \frac{\partial a}{\partial z}$$

Note for now that the second definition and uniformity of the field implies $$div\hat{B}=0$$
in the whole space

Now let's break down the volume that borders our surface into many infinitely small volumes $dV$. From the first definition of divergence, an alternative expression for the flow can be obtained:
$$\Phi = \int_V div \hat{B} dV$$

Because the divergence is zero at any point in space.: $\to$ $$\Phi=0$$

Answer

Q.E.D.