Question

consider a spherical capacitor composed of two concentric shells separated by a perfectly insulating vacuum. the shells are being charged by a current, i, carried by external wires which do not pass through the space between the shells. which of the following statements regarding the region between the shells is true?

there is a charge - carrying current between the shells, but no displacement current between them.
there is both a charge - carrying current and a displacement current between the shells.
there is neither a charge - carrying current nor a displacement current between the shells.
there is a displacement current between the shells, but no charge - carrying current between them.

Explanation:

Step1: Analyze charge - carrying current

Since the region between the shells is a perfect insulating vacuum and the external wires do not pass through this region, there are no free charges (electrons or ions) moving through this region. So, there is no charge - carrying current (conduction current \(I_c=\frac{dq}{dt}\) due to charge motion) in the region between the shells.

Step2: Analyze displacement current

The capacitor is being charged. The electric field \(E\) between the spherical shells is changing with time (\(E=\frac{Q}{4\pi\epsilon_0r^{2}}\), and \(Q\) is changing as the capacitor is charged). The displacement current \(I_d=\epsilon_0\frac{d\Phi_E}{dt}\) (where \(\Phi_E = E\cdot A\) and \(A = 4\pi r^{2}\) for a spherical surface of radius \(r\) between the shells). Since \(E\) is changing, \(\frac{d\Phi_E}{dt}
eq0\), so \(I_d
eq0\)

Answer:

There is a displacement current between the shells, but no charge - carrying current between them.