A Charged Isolated Conductor — Question 2

Question 2

An off-center cavity charge fixes induced totals without fixing their density

An isolated conductor has one closed cavity containing a fixed charge $q=+3.00$ nC, displaced from the cavity center and not touching the wall. The metal itself has net charge $Q_c=-1.00$ nC.

1. Use a Gaussian surface entirely within the metal and surrounding the cavity to find the total inner-wall charge. Then find the outer-surface charge.

2. The charge moves within the cavity without contact. Determine which induced totals stay fixed, and explain whether the inner surface density must remain unchanged.

Question 2 - Solution

An off-center cavity charge fixes induced totals without fixing their density

(a) Method and calculation

$$\mathbf E_{\mathrm{metal}}=0\Rightarrow\Phi_{\mathrm{metal}}=0\Rightarrow q+Q_{\mathrm{inner}}=0.$$$$Q_{\mathrm{inner}}=-3.00\,\mathrm{nC},\quad Q_{\mathrm{outer}}=Q_c-Q_{\mathrm{inner}}=+2.00\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{inner}}\prime=-q=Q_{\mathrm{inner}},\quad Q_{\mathrm{outer}}\prime=Q_c+q=Q_{\mathrm{outer}}.$$$$\int_{\mathrm{inner}}\sigma\,dA=-q.$$The integral is fixed, but local surface density generally redistributes as the cavity charge moves.

Physical check

Charge inside the cavity belongs to the enclosed system but is not part of the specified net charge on the isolated metal.