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We have two equations for the electric field

$$\nabla\times\mathbf{E}=0$$

$$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$$

We also have the boundary conditions

$$E=\frac{\sigma(x,y,z)}{\epsilon_0}\hat{n}$$

At the surface of the cavity.

However, the surface charge ##\sigma(x,y,z)## is an unknown function. Since we do not know a solution for ##\mathbf{E}##, we are unable to determine this surface charge. However, this makes no sense.

If I were to sprinkle a charge density ##\rho(x,y,z)## that is known, surely there can only be one possible surface charge distribution. What equation am I missing?

I would think that it involves the geometry of the conductor outside the cavity, but I'm not sure how to add this in.

Thanks!