Physics 7701: Problem Set #9

Recent changes to this page:

• 07-Nov-2013 --- original version.
• 09-Nov-2013 --- updated problem 3 and 5 comments.

1. Method of images.
In general, follow the guidelines used in class for the problem of a point charge outside a conducting sphere. Make sure you allow for the charge inside to be place at any radius up to a (not just in the center!).
• What does symmetry tell you about the qualitative placement of an image charge? What condition can you use to quantitatively determine where to put it and how large the charge should be? Be sure to check your formula for the potential by evaluating on the surface of the sphere.
• Review the in-class derivation of the surface-charge density.
• Use the image charge!
• In each case, answer about the change (if any) in the potential, induced surface-charge density (including the outside surface), and force. Assume a value for the outer radius (e.g., b). What does superposition say about an added potential V. For a conductor, if there is a charge q in the hollow interior and Q on the sphere, how is that Q distributed between the inner and outer surface? (Remember Gauss's law applied to a Gaussian sphere inside the body of the spherical shell.) Does the image charge change in either of these cases?
2. The potential of a voltage patch.
This is a Dirichlet boundary value problem, which is similar (with a different potential and orientation) of a Green's function example (using the Master formula) considered in class. What is V and what is S? What is the charge density? What is the Dirichlet Green function from the image charge method?
3. Green function expansion.
This is the two-dimensional analog of a problem considered in class. Follow the same steps in each part.
4. Applying a Green function expansion.
Use the Master formula with the result from the last part. What is the charge density to use in this case? (Answer: very simple!) Is there a contribution from the surface term? You will need to integrate over y' from 0 to 1 but you have the y_< and y_> terms. The "trick" is to divide the integral into 0 to y and y to 1. In each of these integrals it is unambiguous what is y and what is y'. Just integrate over y'!
5. The mysterious hollow cube.
One way is to follow the derivation from class for the potential using separation of variables and an expansion in sines, cosines, sinhs, and coshs that we solved for the cube with grounded surfaces but now on the z=0 and z=a surfaces apply the different boundary condition when solving for the Z(z) part. Another way is to use the more general Dirichlet Green function from class (the one with sinh's leads to the same result as the direct expansion of the potential) and apply the master formula. Either way works.
6. More image charges.
In this case the image "charges" are image "lines of charge" but this is completely analogous to using point charges to find the potential in the presence of intersecting grounded planes. Find the surface charge density by relating it to the electric field by Gauss's law and find the field from the scalar potential.
7. Using a cube to simulate a point charge.
Apply the Master formula using the Green function expansion for a rectangular box, generalized to apply to a box centered at the origin. Why is it sufficient to calculate just one face?