Question

Figure 32-20 shows a parallel-plate capacitor and the current in the connecting wires that are discharging the capacitor. Are the directions of (a) electric field $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$and (b) displacement current i_dleftward or rightward between the plates? (c) Is the magnetic field at point P into or out of the page?

Short answer

1. The direction of the electric field$\stackrel{\rightharpoonup }{E}$between the plates is rightward.
2. The direction of displacement current i_d between the plates is leftward.
3. The magnetic field at point P is on the page.

Step-by-step solution

Given

The capacitor is discharging.

Determining the concept

Since the capacitor is discharging, find the direction of the electric field from the direction of the current in the capacitor. Then using the relation between the displacement of the current and electric field, findthe direction of displacement current i_dbetween the plates. Then using the right-hand thumb rule, find the direction ofthemagnetic field at point P.

The formulae are as follows:

$i_d={\epsilon }_{0}A\frac{dE}{dt}$

(a) Determining whether the direction of the electric field E⇀ between the plates is leftward or rightward.

1. While discharging, the current flows away from the positive plate and flows towards the negative plate of the capacitor.
   Hence, in the given figure, the rightward plate is negative, and the leftward is positive.
   The direction of the electric field is always positive to negative.
   Therefore, the direction of the electric field $\stackrel{\rightharpoonup }{E}$between the plates is rightward.

(b) Determining whether the direction of displacement current id between the plates is leftward or rightward

(b). The displacement current in the capacitor is given by

$i_d={\epsilon }_{0}A\frac{dE}{dt}$

While discharging,theelectric field between the plates decreases. Hence,$\frac{dE}{dt}$is negative. Thus, the displacement current is negative. It implies that its direction is opposite to the electric field.

Therefore, the direction of displacement current i_d between the plates is leftward.

(c) Determining whether the magnetic field at point P is into or out of the page

(c). Since the current is flowing leftward, the magnetic field at point P is into the page according to the right-hand thumb rule.
The direction of the electric field between the plates depends on whether the capacitor is charging or discharging, and the direction of the displacement current depends on the direction of the electric field.