Gauss' Law for Magnetic Fields — Question 9

Question 9

Zero divergence does not require zero circulation

A static field in a region is $\mathbf B=k(-y\hat x+x\hat y)$, where k is constant. Use $\oint\mathbf B\cdot d\mathbf l=\mu_0I_{\mathrm{enc}}$ for a circle in the xy-plane.

1. Check its divergence and find its circulation around radius a, counterclockwise from +z.

2. Infer the uniform axial current density compatible with this field.

Question 9 - Solution

Zero divergence does not require zero circulation

(a) Method and calculation

$$\nabla\cdot\mathbf B=\partial_x(-ky)+\partial_y(kx)=0.$$$$\boxed{\oint\mathbf B\cdot d\mathbf l=(ka)(2\pi a)=2\pi ka^2}.$$

(b) Interpretation and further calculation

$$\mu_0J_z\pi a^2=2\pi ka^2.$$$$\boxed{J_z=2k/\mu_0}.$$

Physical check

Divergence governs closed flux, whereas curl and circulation concern a different geometric property.