- Problem 1 [10 points]
Prove this mean value theorem: For charge-free space in the electrostatic limit, the value of the electrostatic potential φ at any point in space is equal to the average of the potential over the surface of any sphere centered on that point.
Hint: Use the fact that where there are no charges ∇2φ=0. Then relate the averge φ over the surface of a shpere to the flux of electric field through the surface.
Functions which satisfy Laplace's equation are called harmonic functions; harmonic functions obey the above mean value theorem (in other words, you have to prove this mean value theorem, you cannot simply say it is true because harmonic functions obey it!).
- Problem 2 [10 points]
Answer Discussion Questions 2.2.1 and 2.2.2 in Notes 2-2.
- Problem 3 [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 φo? If the sphere has a fixed total charge Q?
- Problem 4 [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