Potential of a Charged Isolated Conductor — Question 7

Question 7

Grounding a cavity shell removes its exterior charge but not its interior field

A spherical conducting shell with inner radius a and outer radius b encloses a centered point charge q. The shell is connected to a distant zero-potential reservoir. There are no external sources.

1. Find both surface charges and the net charge supplied to an initially neutral shell.

2. Find V in the cavity and exterior, and compare the electric work on a probe q0 between two cavity radii r1 and r2.

Question 7 - Solution

Grounding a cavity shell removes its exterior charge but not its interior field

(a) Method and calculation

$$Q_{\mathrm{inner}}=-q,\quad V_{\mathrm{metal}}=kQ_{\mathrm{outer}}/b=0\Rightarrow Q_{\mathrm{outer}}=0.$$$$Q_{\mathrm{shell}}=-q,\quad\Delta Q_{\mathrm{reservoir\to shell}}=-q.$$

(b) Interpretation and further calculation

$$V(r)=kq(1/r-1/a)\ (0<r<a),\quad V(r)=0\ (r\ge a).$$$$W_e=q_0[V(r_1)-V(r_2)]=kqq_0(1/r_1-1/r_2).$$

Physical check

Grounding fixes the conductor voltage; it does not remove the field generated by a charge inside its cavity.