Gauss' Law for Magnetic Fields — Question 5

Question 5

Solenoidality reconstructs a missing transverse component

In a region, $B_x=\alpha x^2y$, $B_z=0$, and $B_y(x,0,z)=0$. The constant alpha carries the units required for B to be in teslas.

1. Find By using $\nabla\cdot\mathbf B=0$.

2. Verify the divergence of the reconstructed field and state why the boundary value is needed.

Question 5 - Solution

Solenoidality reconstructs a missing transverse component

(a) Method and calculation

$$\partial_xB_x=2\alpha xy\Rightarrow\partial_yB_y=-2\alpha xy.$$$$B_y=-\alpha xy^2+f(x,z),\quad B_y(x,0,z)=0\Rightarrow\boxed{B_y=-\alpha xy^2}.$$

(b) Interpretation and further calculation

$$\nabla\cdot\mathbf B=2\alpha xy-2\alpha xy+0=0.$$$$\boxed{\text{Without boundary data, any }f(x,z)\text{ remains undetermined.}}$$

Physical check

The divergence equation constrains a derivative, so an integration function survives until boundary data fix it.