Something occured to me just now. A question about the scalar potential.(adsbygoogle = window.adsbygoogle || []).push({});

First I will do some calculations of the laplacian of the scalar potential in different electrostatic situations to give myself a basis for my question.

Point charge:

[tex]\phi =\frac{1}{4\pi\epsilon_0} \frac{q}{r}[/tex]

[tex]\nabla^2\phi=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \phi}{\partial r}\right)=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\left(-\frac{1}{4\pi\epsilon_0} \frac{q}{r^2}\right)\right)=\frac{1}{r^2}\frac{\partial}{\partial r}\frac{q}{4\pi\epsilon_0}=0[/tex]

Line of uniform charge density:

[tex]\phi=\frac{1}{2\pi\epsilon_0}\lambda \ln r[/tex]

[tex]\nabla^2\phi=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \phi}{\partial r}\right)=\frac{1}{r}\frac{\partial}{\partial r}\left(r\left(\frac{1}{2\pi\epsilon_0}\lambda\frac{1}{r}\right)\right)=\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{1}{2\pi\epsilon_0}\lambda\right)=0[/tex]

Plane of uniform charge density:

[tex]\phi=\frac{z\sigma}{2\epsilon_0}[/tex]

[tex]\nabla^2\phi=\frac{\partial^2\phi}{\partial z^2}=0[/tex]

My question then arises: Is the laplacian of the scalar potential always zero?

If no, please show a counterexample.

Thanks.

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# Laplacian of electrostatic potensial

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