| For Problem 2: Is Sigma the wire's Linear Charge Density (i.e., in units of Coloumb/meter) ? | Yes! |
| Regarding Problem 1: Is phi(x,y) on the z=0 plane an exact solution? Can we just design some conductor shapes to approximately generate this potential? | It should be an exact solution. If you can only find an approximate solution, please write it down for partial credit. |
| I am simply clueless for #2 because I only have method of images in my mind, which I have assured myself will not work. | Hint: try to get ordinary differential equations for the fields as a function of z. |
| Another question on #1; I think your expression for phi needs to be revised. Here's my argument: because the potential does not depend on y, the conductors should be infinitely long in the y-direction. However, if this is the case, the potential should depend on something like Ln(1/something), not just 1/something. If we naively use 2-D to examine the problem, we will of course get 1/something. But by that we are neglecting the contribution from the "infinitely long conductor". Please let me know what you think. | I assume that by "something" you mean a "polynomial in x,z." Ln(1/r^2) is indeed the potential around an infinitely thin wire (up to a constant), but a nonsingular configuration of charges does not have to produce Ln(polynomial). |
| Problem 2: When you say the power per unit length, is that length along the direction of motion, or along the length of the wire? Also, do you want the energy radiated per x, or the power (i.e., the time derivative)? | The "length" is along the wire. So "power per unit length" means dE/dy dt. You may calculate (dE/dy dx) instead, and let me know how to relate it to (dE/dy dt). |
| I am having some trouble getting started with problem #1. The problem seems as though it should be approached via trial-and-error of the conductor shapes. Is this what you expect us to do or is there a more methodical approach. | There is indeed a more methodical approach. I think it would be very hard to find the answer by trial-and-error. Perhaps it would help to mention that there are two steps to the solution. |
| Can you clarify at all what you meant by your hint to "try to get ordinary differential equations for the fields as a function of z?" | Well, you'll have to solve some differential equations, and the hint says that with the right techniques the differential equations can be reduced to differential equations in z only. (This does not necessarily mean that the fields depend only on z, but that the equations can be reduced to ODEs in z.) I hope that helps. |