Consider a grounded, conducting, spherical shell of outer radius b and inner radius a. Using the method of images, dicuss the problem of a point charge q inside the shell, i.e. at a distance r<a from the center. Find a) the potential inside the sphere; b) the induced surface charge density on the inner surface of the shell at r=a; what is the total induced charge? c) the magnitude and direction of the force acting on q. Does q get pushed towards the center, or away from the center? d) Is there any change in the solution if the sphere is kept at a fixed potential φo? If the sphere has a fixed total charge Q?

Consider the four point charges shown in the figure on the right, arranged on the edges of a square of side a.a) Find the value of Q (in terms of q), such that the electric field at the center of the square points down (i.e., it only has a –y component). Show all work and justify any assumptions you make.b) Assuming Q has the value found in part (a), find the force exerted by the four corner charges on a fifth charge −q placed at the center of the square.c) Also using the result from (a), find the electric field at point P (shown in the figure) created by the four corner charges (there is no charge at the center of the square for this part).

Figure 1 shows a configuration of a flat plane and an arbitrary shape, both carrying surfacecharge density of magnitude σ. The arrows show the electric field lines. Are the statementsbelow true or false?a) The flat plane is carrying a negative surface charge density −σ and the arbitrary shape iscarrying a positive surface charge density +σ. Justify your answer.b) Both surfaces are equipotential surfaces. Justify your answer.

1. Consider the four point charges shown in the figure on the right, arranged on the edges of a square of side a. a) Find the value of Q (in terms of g), such that the electric field at the center of the square points down (i.e., it only has a -y component). Show all work and justify any assumptions you -29 a a/2 make. b) Assuming Q has the value found in part (a), find the force exerted by the four corner charges on a fifth charge -q placed at the center of the square. c) Also using the result from (a), find the electric field at point P (shown in the figure) created by the four corner charges (there is no charge at the center of the square for this part).

Suppose you design an apparatus in which a uniformly charged disk of radius R is to produce an electric field. The field magnitude is most important along the central perpendicular axis of the disk, at a point P at distance 2.60R from the disk (see Figure (a)). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius R/2.60 (see Figure (b)). Assume that the ring will have the same surface charge density as the original disk. If you switch to the ring, by what part will you decrease the electric field magnitude at P? Number i Units (a) (b)

A thin metal ring of radius R is placed horizontally and charged with charge Q. A point like object with mass m and charge q is let fall vertically from a height H measured from the plane of the ring. The initial velocity is zero. The object is falling along the axis of a ring. What is the velocity of the object, when it reaches the middle of the ring? Gravitational acceleration is denoted by g.