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The picture is a question from the book Intro to Electrodynamics by Griffiths. In question as you can see we want to find potential due to an infinite strip maintained at constant potential in the given region (slot).

My question is that the fourth boundary condition they used is ($V\to0$ as $x\to\infty$) which I think is wrong . Because when we deal with infinite charge distribution (like infinite line of charge or infinite sheet of charge) the convention that potential zero at infinity fails. For infinite charge distribution potential at infinity blows up. And hence we cannot use ($V\to0$ as $x\to\infty$) condition. So what is wrong with my argument here?

Kyle Kanos
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Vicky Shrimali
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2 Answers2

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The Answer is in Gauss's law. The Electric field for sheet is given by $$E = \frac{\sigma}{\epsilon_0}$$ Here $\sigma = Q/A$, where $Q=$ charge and $A=$ area of the sheet. So, In this case charge $Q$ and area $A$ both tends to infinity. Which results into finite value Electric field $E$. And, as $E$ is proportional to potential $V$, potential is also finite.

Divyang Prajapati
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The potential $V$ between the parallel planes at potential zero decreases exponentially to zero for $x\to\infty$ due to the shielding of the parallel metal plates at $V=0$. In this case the requirement$V\to0$ as $x\to\infty$ is actually not necessary. It ensures only that there is no infinite charge at infinity which would produce an potential solution which increases exponentially with $x$. In z-direction the potential is constant.

freecharly
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