Question

A charge q is a distance b > R from the center of a grounded conducting sphere (that is, a sphere at zero potential, perhaps connected by a thin wire to an enclosing spherical shell very, very far away) of radius R. We want to find an image charge q′ at a distance a < R from the center of the sphere along the same line (take it to be the z-axis) as q such that the potential due to q and q′ at r = R is zero. Clearly, q′ must be negative and proportional to q (q′ = −αq).

(a) Calculate the potential φ at an arbitrary point from q and q′ in terms of r, θ, q, and α.

(b) Set the potential from part (a) to zero at r = R. This gives an equation with two unknowns, α and a. But this equation must be satisfied for all θ, which allows you to determine both unknowns. [Hint: You should have two terms in the potential. Get one on each side of the equals side, invert, and square. This should give you an equation of the form A + Bcosθ = A′+ B′cosθ. Since this must be equal for all θ,we must have A = A′ and B = B′. Getting α in terms of a and b should be straight forward. Then you should have a quadratic equation in a. One of the solutions is trivial, and you should ignore it. It is the other solution that we want.]

(c) From the potential in part (a) with the parameters you found in part (b), calcu- late the electric field outside the shell from E⃗ = −∇⃗ φ.

(d) From the electric field just outside the surface of the sphere, calculate the surface charge density σ on the sphere.

(e) Integrate σ from part (d) over the surface of the sphere to obtain the charge on the sphere. Where did this charge come from?

Answer 1

The solution is given in image files attached below with complete solution.