Why does the method of images work?

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The method of images is effective in electrostatics for calculating electric fields near conducting surfaces by using imaginary charges to satisfy boundary conditions. It relies on the uniqueness theorem, which states that if a solution for electric potential meets the boundary conditions and charge density, it is the only solution. This method allows for the clever arrangement of charges outside the region of interest to fulfill these conditions, often utilizing symmetries. By applying this technique, one can simplify complex problems in electrostatics. Ultimately, the method of images provides a unique and efficient way to solve for electric potentials in specific configurations.
Shreyas Shree
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Why does the method of images work?
 
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Which method ?
 
Method of image charges in electrostatics to calculate the distribution of the electric field of a charge in the vicinity of a conducting surface.
 
Have you read the relevant Wikipedia articles?

https://en.wikipedia.org/wiki/Method_of_image_charges
https://en.wikipedia.org/wiki/Uniqueness_theorem_for_Poisson's_equation

In short, if you can find a solution for the electric potential that satisfies your boundary conditions for the electric potential, and the charge density within the considered region, you've found the unique solution. This is due to the uniqueness theorem.

That fact allows you to cleverly pick an arrangement of charges outside your considered region that satisfies your electric potential boundary conditions (usually by taking advantage of/introducing symmetries) and know that you've found the only solution that exists for your configuration.
 
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