Constant magnetic field > Magnetic flux > Problem 9.4.4

Statement

$9.4.4.$ Prove that the magnetic flux generated by the current element through any closed surface is zero.

Solution

In the previous problem, we already showed that if the field divergence is zero, then its flow through any closed surface is zero.

Let's write down the Biot-Savare-Laplace law:

$$\hat{B} = \mu_0 I[d\hat{l} \times \frac{\hat{r}}{r^3}]$$

Note that the gradient from the following expression is:
$$\nabla\frac{1}{r} = -\frac{\hat{r}}{r^3}$$

Now let's find the divergence:
$$div \hat{B} =-\nabla\mu_0 I[d\hat{l} \times \nabla \frac{1}{r}]$$

Only $r$ depends on $x, y, z$, here we need one fact from mathematics, the rotor from the gradient is 0 $\to$ divergence is zero, which means the statement is proven.

Answer

Q.E.D