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I am learning how to solve Laplace and Poisson equations and, usually, when I am solving the exercises and I need to set the boundary conditions, I use this and it always works:

$$\frac {\partial \varphi}{\partial n} = 0$$

But I would like an intuitive explanation to this derivative, how to know that the variation of the electric potential ($\varphi$) over the normal vector ($\vec{n}$) is 0? Is there any example in which $\frac {\partial \varphi}{\partial n} \neq 0$?

Thanks!!

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Giving the normal derivative on the boundary is the Neumann boundary condition. The normal derivative of the potential is the normal component of the electric field. This boundary condition means that the surface charge density is zero everywhere on the boundary and, according to Gauss law, you have no net total charge in the interior of the boundary.

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  • $\begingroup$ Minor wrinkle: Carl Neumann, not 'Carl von Neumann'. $\endgroup$
    – Gert
    Sep 24, 2016 at 20:51

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