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In a problem with a sphere and a charge $q$ at a distance $l$ away from it, the solution involves putting the image charge inside the sphere at $a = \frac{R^2}{l}$. But doesn't that contradict the fact that we shouldn't put an image charge in volume $V$ of the domain where the boundary of the domain is the surface of the sphere? Our domain is the points inside the sphere and we aren't allowed to put image charge there.

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You want to determine the electric field and electric potential outside the sphere. Therefore, the domain $V$ is the region outside the sphere. The surface of the sphere is the boundary of this domain, and the region inside the sphere is not part of this domain. So it's perfectly fine to place an image charge inside the sphere, since this is outside the boundaries of the domain of interest.

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