1-D Cartesian Green Function Two infinite and grounded conducting sheets are in the x = 0 and 2 = w plane. In the x = x'

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1 D Cartesian Green Function Two Infinite And Grounded Conducting Sheets Are In The X 0 And 2 W Plane In The X X 1 (285.71 KiB) Viewed 35 times

1-D Cartesian Green Function Two infinite and grounded conducting sheets are in the x = 0 and 2 = w plane. In the x = x' plane, there is an infinite non-conducting sheet with surface charge density o'. 1. Find the potential, tr(), on the left (0<x<') and to the right (z' <x<w), Ur(2), of the non-conducting sheet using any method (Gauss's law or the boundary value method can be used: you should be able to do it using both methods, but you need to only show your work using one method). 2. Write the potential v(2) for 0<x< was a single function using y, and y, and the Heavyside step function e. (In the future this (with o'/€, set to 1), will be called a Green function, which is the motivation for the title of this problem.) 3. Show that V2v(x) = 28(x - x'). You will need to use the fact that do(x)/dx = 8(x), and d0 (-x)/dx = -8(x). Also compute 524(), where the prime means to take derivatives with respect to primed variables. (This may seem odd because ' was defined to be a constant; here you are being asked to treat it as a variable. You should get an answer that is proportional to 8(x - 2')). As discussed in class, the motivation for solving this problem is that its potential, y, can be used in Green's second identity (eqation 1.35), which is a form of reciprocity, to solve the most general problem for this geometry. The most general problem is to find the potential (x) when p = p(x) between x = a and x = b. the left plane is grounded, and the right plane is at potential V. Instead of using y found above to solve the most general problem, first use it to solve an easier problem: 4. Use Equation 1.35 and (x) to find the potential (2) when p(x) = 0 between the conductors and (0) = 0 and (w) = V. Include a sketch or a sentence where you define V and S when you use Equation 1.35. There will be a subtilty with notation here - if you use Equation 1.35 as written, you'll end up with (2') and not the desired (2).