A point charge \(Q\) is placed a distance \(R\) from the center of a conducting
sphere of radius \(a,\) with \(R>a\) (the point charge is outside the sphere). The
sphere is grounded, that is, connected to a distant, unlimited source and/or
sink of charge at zero potential. (Neither the distant ground nor the
connection directly affects the electric field in the vicinity of the charge
and sphere.) As a result, the sphere acquires a charge opposite in sign to
\(Q\), and the point charge experiences an attractive force toward the sphere.
a) Remarkably, the electric field outside the sphere is the same as would be
produced by the point charge \(Q\) plus an imaginary mirror-image point charge
\(q\), with magnitude and location that make the set of points corresponding to
the surface of the sphere an equipotential of potential zero. That is, the
imaginary point charge produces the same field contribution outside the sphere
as the actual surface charge on the sphere. Calculate the value and location
of \(q\). (Hint:
By symmetry, \(q\) must lie somewhere on the axis that passes through the center
of the sphere and the location of \(Q .)\)
b) Calculate the force exerted on point charge \(Q\) and directed toward the
sphere, in terms of the original quantities \(Q, R,\) and \(a\)
c) Determine the actual nonuniform surface charge distribution on the
conducting sphere.
