Let us assume a finite, conducting plate of dimension: $10\mathrm{m} \times 10\mathrm{m} \times 1\mathrm{m}$. I want to determine the electric field at the middle of one of the plates $10\mathrm{m} \times 10\mathrm{m}$ surfaces. Using Gauss's law one finds the electric field to be:
$$E= \frac{\rho}{\epsilon_0}$$ and we see that the electric field is not depend on the distance from the surface. I know that's the solution for a infinite plate. For a finite plate that doesn't seem very realistic. I assume the electric field to be not orthogonal onto the surface but to diverge - am I right with that assumption? Somehow the field has to decrease with distance. How do I modify my approach with Gauss's law so that I find the right solution for a FINITE plate?
Answer
You can't do this problem with Gauss's law, because you don't have the symmetry needed to assume the direction of the electric field. You have to break the square down into differential bits with area $dxdy$, and then integrate coulomb's law.
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