Displacement Current — Question 2

Question 2

A smaller contour encloses only a fraction of uniform displacement current

A circular vacuum capacitor of plate radius a is charged with current I. The gap field is uniform in the quasistatic approximation. Consider a centered circular contour of radius r<a.

1. Find the displacement current enclosed by that contour and its magnetic field.

2. At r=a/3, compare enclosed current and B with their edge values.

Question 2 - Solution

A smaller contour encloses only a fraction of uniform displacement current

(a) Method and calculation

$$J_d=I/(\pi a^2),\quad I_{d,\mathrm{enc}}=J_d\pi r^2=I r^2/a^2.$$$$\boxed{B_\phi=\mu_0Ir/(2\pi a^2)}.$$

(b) Interpretation and further calculation

$$r=a/3\Rightarrow I_{d,\mathrm{enc}}=I/9.$$$$\boxed{B(a/3)=B(a)/3}.$$

Physical check

A contour-area fraction determines enclosed current, but the circumference contributes another factor of radius to B.