# Home Work Set # 6, Physics 217, Due: October 24, 2001

### Problem 1

A uniform line charge λ is placed on an infinite straight wire, a distance d above a grounded conducting plane. The wire runs parallel to the x axis and directly above it, and the conducting plane is the xy plane.
a) Find the potential in the region above the plane.
b) Find the charge density σ induced on the conducting plane.

### Problem 2

In Example 3.2 of Griffiths we assumed that the conducting sphere was grounded (V = 0). But with the addition of a second image charge, the same model will handle the case of a sphere at any given potential V0 (relative, of course, to zero at infinity).
a) What charge should you use, and where should you put it?
b) Find the force between a point charge q and a conducting sphere at potential V0.

### Problem 3

Two infinite parallel grounded conducting planes are held a distance a apart. A point charge q is placed in the region between them, a distance x from one plate.
a) Find the force on q.
b) Check that your answer is correct for the special case in which a → ∞. Check that your answer is correct for the special case in which x = a/2.

### Problem 4

A long rectangular pipe, running parallel to the z axis, has three grounded metal sides, at y = 0, y = π, and x = 0. The fourth side, at x = a, is maintained at a specified potential V0(y).
a) Develop a general formula for the potential within the pipe.
b) Find the potential explicitly, for the case V0(y) = V0 = constant.

### Problem 5

A cubical box consists of five metal sides (length of each side is π) which are welded together and grounded (see Figure 1). The top is made of a separate sheet of metal, insulated from the rest, and held at a constant potential V0 by a battery. Find the potential inside the box.

Figure 1. Problem 5.