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My lecturer and I have found separately valid solutions to Poisson's equation in the region of interest for the following problem:

enter image description here

Here is my interpretation of the boundary conditions:

$$V(x,y,z \to \infty) = 0$$ $$V(x,y,z = \sqrt{a^2 - x^2 - y^2})=0$$ $$V(x,y,0) = 0$$

With this, I could find a solution which satisfies all three conditions:

$$V = \frac{z^2 - z \sqrt{a^2 -x^2 -y^2}}{e^z}$$

And thus I can invoke the uniqueness theorem to state that this is the only solution. However, my lecturer used three separate images, and shown that the solution was:

enter image description here

Which also seems to indeed satisfy the boundary conditions (however his differed from mine with $V(r=a)=0$ which I disagree with as I believe it should be $V(r \le a) = 0$ as the potential should be $0$ everywhere inside the conductor.

How can we have simultaneously the unique solution? Why is my answer incorrect? How am I supposed to figure out the images to use for this problem? How can I reliably use the method of images if I can't eyeball the correct images to use?

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  • $\begingroup$ Your solution $V$ does not have any dependency on the point charge $q$ and the distance $d$. $\endgroup$ – K_inverse Mar 20 at 0:35
  • $\begingroup$ I don't think $V(x,y,0)=0$ for $\sqrt{x^2+y^2}<a$. The hemisphere is a bump, not a solid conductor. $\endgroup$ – Aaron Stevens Mar 20 at 5:09
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Your $V$ doesn’t depend on $q$ and $d$ so it can’t satisfy Poisson’s equation. It doesn’t seem to even satisfy Laplace’s equation. So you do not both have the unique solution.

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