(adsbygoogle = window.adsbygoogle || []).push({}); 1. This is a 2D Laplace eqn problem. A semi-infinite strip of width a has a conductor held at potential [tex] V(0,a) = V_0 [/tex] at one end and grounded conductors at y=0 and y=a. Find the induced surface charge [tex] \sigma (y) [/tex] on the conductor at x=0.

2. Relevant equations.

The potential is

[tex] V(x,y) = \frac{4V_0}{\pi}\sum_{n=0}^{\infty}\frac{e^{-(2n+1)\pi x / a}}{2n+1} \sin(\frac{(2n+1)\pi y}{a} ) [/tex].

3. I tried

[tex] \sigma = \epsilon_0 E_{norm} = -\epsilon_0 \left . \frac{\partial V}{\partial x} \right |_{x=0} [/tex]

which yielded the bounded but non-convergent series

[tex] \sigma(y) = \frac{4V_0\epsilon_0}{a} \sum_{n=0}^{\infty} \sin ( \frac{(2n+1)\pi y}{a} ) [/tex].

Differentiation with respect to x killed the 1/(2n+1) factor and then evaluating at x=0 killed the exponential, leaving nothing to cause the terms to decay with higher n. It makes me doubt that using the normal derivative of the potential is the right way to get the field, but everything I've read on it states with demonstration that the field at the surface of a conductor is [tex] - \epsilon_0 \partial V / \partial n [/tex]. So what to do?

Thanks for reading!

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# Homework Help: Non-convergent series for surface charge density

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