Problem Set 3

Due Thursday, September 24, by 5pm

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Problem 1 [10 points]
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 φ_o? If the sphere has a fixed total charge Q?
Problem 2 [10 points]
A line charge λ is placed parallel to, and a distance R away from, the axis of a conducting cylinder of radius b that is held at fixed voltage such that the potential vanishes at infinity. Find
a) the magnitude and position of the image charge(s)
b) the potential at any point (expressed in polar coordinates with the origin at the axis of the cylinder and the direction from the origin to the line charge as the x axis)
c) the induced surface charge density, and plot it as a function of angle for R/b = 2 and 4 in units of λ/2πb
d) the force on the line charge per unit length