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Charge within a cavity inside a conducting material

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Suppose I have a charge inside a conductor as shown in the image I've attached. For any charge distribution: [tex] \oint \mathbf{E} \cdot d \mathbf{a} = 0 [/tex]

I can see that if I took some path from the charge, through the conductor, and back to the charge, the integral would be zero still.

Now, that charge is in the center of the cavity, but now imagine the charge is shifted to the right by some amount. Let's say I took a path clockwise by going straight left from the charge, then through the conductor, then came out of the conductor to go straight left again to meet the charge. Inside the conductor [itex] \mathbf{E} = 0 [/itex] so that doesn't contribute to the integral. But the path to the left leaving the charge is longer than the path on the right that comes back to the charge. This seems to give a nonzero integral which is a violation.

I know the charge would shift the charge on the surface of the cavity, but the E fields due to those charges would cancel since the total E field inside the cavity would be zero without the charge on the inside. I'm not sure how the integral would equal zero then.

Attached Images
File Type: png charge in a cavity.PNG (17.0 KB)

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