A Charged Isolated Conductor — Question 8

Question 8

Nested conducting shells transmit an enclosed-charge constraint through the gaps

A charge q is centered within two concentric conducting shells. Their radial intervals are $a<r<b$ and $c<r<d$, with $a<b<c<d$, and net charges are $Q_A$ and $Q_B$. There are no external sources.

1. Find the charges on all four conductor surfaces.

2. Find the field in $b<r<c$ and $r>d$. Determine the choice of $Q_B$ that makes the exterior field vanish, and state whether that also forces the gap field to vanish.

Question 8 - Solution

Nested conducting shells transmit an enclosed-charge constraint through the gaps

(a) Method and calculation

$$Q_a=-q,\quad Q_b=Q_A+q.$$$$Q_c=-(q+Q_A),\quad Q_d=Q_B+q+Q_A.$$

(b) Interpretation and further calculation

$$E_r=\frac{q+Q_A}{4\pi\epsilon_0r^2}\quad(b<r<c),\quad E_r=\frac{q+Q_A+Q_B}{4\pi\epsilon_0r^2}\quad(r>d).$$$$Q_B=-(q+Q_A)\Rightarrow E_{\mathrm{exterior}}=0.$$The gap field remains nonzero unless $Q_A=-q$ independently.

Physical check

Each metal region has zero field, but each vacuum gap responds to the total charge enclosed within that gap.