Gauss' Law for Magnetic Fields — Question 4

Question 4

A linear three-dimensional field must have zero sum of normal gradients

A proposed field is $\mathbf B=(ax,by,cz)$ in a rectangular region, where a, b, and c are constants with units T/m.

1. Use a small box to find the constraint imposed by zero closed magnetic flux.

2. For a=.20 T/m and b=-.05 T/m, find c and the flux through a unit-area z-face at z=2.00 m with normal +z.

Question 4 - Solution

A linear three-dimensional field must have zero sum of normal gradients

(a) Method and calculation

$$\Phi_x=a\Delta x\Delta y\Delta z,\quad\Phi_y=b\Delta x\Delta y\Delta z,\quad\Phi_z=c\Delta x\Delta y\Delta z.$$$$\boxed{a+b+c=0}.$$

(b) Interpretation and further calculation

$$c=-a-b=-.150\,T/m,\quad B_z(2)=-.300\,T.$$$$\boxed{\Phi_{+z}=-.300\,Wb\quad(A=1.00\,m^2)}.$$

Physical check

A nonzero gradient in one direction must be balanced by the other directions.