January 25, 2009

PHY 712 - Problem Set #4

Continue reading Chaper 1 & 2 in Jackson; homework is due Wednesday, Jan. 28, 2009.

1. Consider a one-dimensional charge distribution of the form:
   
   $$\rho(x) = \left\{ \begin{array}{lll} 0 \;\; & {\rm {for}}\; ... ...\\ 0 \;\;\; & {\rm {for}} \; & x \geq a, \end{array} \right.$$
   
   
   where $\rho_0$ and $a$ are constants.
   1. Solve the Poisson equation for the electrostatic potential $\Phi(x)$ with the boundary conditions $\Phi(0) = 0$ and $\frac{d \Phi}{dx}(0) = 0$.
      1. Use the Green's function discussed in Lecture Notes #4:

         
         

         $$G(x,x') = 4 \pi x_<.$$
         
         

      2. Use the Green's function discussed in Lecture Notes #5:

         
         

         $$G(x,x') = \frac{8 \pi}{a} \sum_n \frac{\sin(n\pi x/a)\sin(n\pi x'/a)}{\left(\frac{n\pi}{a}\right)^2}.$$
         
         

   2. In both cases, check whether the Green's function-derived solutions satisfy the boundary conditions. If they do not, you will need to add contributions from solutions to the homogeneous equations as discussed in Lecture Notes #5. Obviously, you should obtain the same answer for both methods.

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