A neutral conductor contains a hollow cavity in which there is a +90.0 nC point charge. A charged rod then transfers -30.0 nC to the conductor. Afterward, what is the charge. . .
. . . on the inner wall of the cavity?
. . . on the exterior surface of the conductor?
Solution
Let’s call the charge on the inner wall Qin, the charge on the outer wall Qout, the total
charge of the conductor Qc, and the charge of the point-charge Qp.
To begin, recall two special properties of a conductor: its charge is distributed over its
surface(s), and the electric field inside it is zero.
The first tells us that Qc = Qout + Qin.
The second implies that the electric flux, Φe, through any Gaussian surface that is
entirely within a conducting material is also necessarily zero. Suppose we consider such
a surface surrounding the cavity. As noted, because the surface is entirely within the
conducting material, Φe = 0. By Gauss’s law, however, we know Φe = Qenc/0, where
Qenc = Qin + Qp is the total charge contained within the surface. Hence, we have
0 = Qin + Qp, or Qin = −Qp.
Together, these give
Qin = −Qp
Qout = Qc − Qin = Qc + Qp.
Given Qc = −50 nC and Qp = 100 nC, this gives Qin = −100 nC and Qout = 50 nC.