A parallel‑plate capacitor moves with velocity $v$, as shown in the figure. The electric field strength between the plates is $E$. Determine the rate of change of the electric flux through the rectangular contour $abcd$ and the circulation of the magnetic induction around this contour. How are the sought quantities related to each other in SI? In CGS?
$b.$ Give examples confirming the proportionality of the circulation of the magnetic induction around a contour to the rate of change of the electric flux through the surface bounded by this contour.
Solution
$a.$$$\frac{dN}{dt} = E \frac{dS}{dt} = v l E.$$
Choose a surface passing through the capacitor:
$$C_B = \int \vec B \\, d\vec l = \mu_0 \sum I = \mu_0 \frac{\sigma l \cdot v \\, dt}{dt} = \mu_0 \varepsilon_0 v l E.$$
Using the solution of problem 11.6.1, this can be generalised for any surface:
$$C_B = \mu_0 \varepsilon_0 \frac{dN}{dt} \quad \text{(in SI)}, \qquad C_B = \frac{1}{c} \frac{dN}{dt} \quad \text{(in CGS)}.$$$b.$
Charging capacitor:
When a capacitor is charging, current flows in the wires, but there is no current between the plates. The changing electric field between the plates maintains the circulation of $B$ around a contour that does not intersect the conductor, equal to the circulation around a contour that does intersect the conductor: $C_B = \mu_0 I$ for a surface crossing the conductor.
Answer
$$\boxed { a. \ \frac{dN}{dt} = v l E, \quad C_B = \mu_0 \varepsilon_0 v l E, \quad C_B = \mu_0 \varepsilon_0 \frac{dN}{dt} \ \text{(in SI)}, \quad C_B = \frac{1}{c} \frac{dN}{dt} \ \text{(in CGS)}. }$$
