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Say I'd like to find the potential created with a conducting sphere and an external point charge. When using the method of images to find the potential, we know that that potential is unique for the region of interest (outside the sphere), as long as it is defined on all boundaries and as long as the charge density is specified in the region.

If I wanted to find the surface charge density of the conductor, using the potential $V$ arrived at via the method of images, I can do so with

$$ \sigma=\frac{1}{4\pi}E_r = -\frac{1}{4\pi}\frac{\partial V}{\partial r}. $$

From a mathematical point of view, this makes sense. But how can a collection of point charges recreate a continuous charge distribution? The method of images emphasizes the uniqueness of the solutions, and I think that because $V$ is uniquely determined then $\rho$ must be as well. But I cannot wrap my head around how 3 point charges could create a unique, continuous charge density. I would think it should be some collection of delta functions.

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But how can a collection of point charges recreate a continuous charge distribution? The method of images emphasizes the uniqueness of the solutions, and I think that because $V$ is uniquely determined then $\rho$ must be as well.

The reasoning is a bit more complex than that. Uniqueness tells us that if you specify the potential on a surface, and demand that there is no charge outside the surface, then the potential outside the surface is uniquely determined. And indeed, both the actual charge distribution and the image charges produce the same potential outside, even though they have different charge distributions and potentials inside.

Now, using the image charge configuration, you can easily determine the potential outside. So to get the actual charge distribution, you additionally impose the condition that the potential is uniform inside. With this additional condition, you now have the potential everywhere, which determines the charge distribution everywhere. In other words, all the image charge trick is doing is giving you an easy way to compute the potential outside, which is the part that's actually unique.

Incidentally, the fact that a charge distribution inside a surface isn't determined uniquely by boundary conditions on a surface should be very familiar. The shell theorem says that the potentials outside a uniformly charged spherical shell, and a point charge at the center, are identical.

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  • $\begingroup$ Hm, I may be misunderstanding the boundary definition then. The image charges are obviously firmly within the sphere, i.e. outside of the area of interest, but isn't the surface charge of the sphere on the boundary, i.e. within the area of interest? If the boundary is included, wouldn't the area of interest have different charges for each case; a point charge + surface charge for one, and a lone point charge for the other? $\endgroup$ Oct 5, 2020 at 19:26
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    $\begingroup$ @AlexWofford The potential is fixed on the surface, and unique outside the surface. But the charge density depends on how the potential's slope changes across the surface, which means it depends on the potential inside the surface, which means it's not uniquely determined. $\endgroup$
    – knzhou
    Oct 5, 2020 at 19:33

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