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How can I calculate the force on a charged particle due to an uncharged infinite conducting plate?

If there is a small object with positive charge placed above a metal plate, the object induces a negative charge on the surface of the plate facing the object. Let's call this surface as $S_1$.

Since the conductor was uncharged initially, and charges always reside on the surface of a conductor, $S_2$ gets a positive charge.

enter image description here

From the picture above, the particle would experience an upward force due to $S_2$ and downward force due to $S_1$. So is the force zero?

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This the kind of question that can be solved by the method of images.

Try placing a fictitious charge on the other side on the conducting plane. You should arrange it in such a way that the electrostatic potential is precisely zero on the surface of the conductor. If your case you put it at equal distance as the first but on the other side.

The physical interpretation is that the electrons in the neutral conductor will rearrange themselves because of the field from $q$. Since the conductor is infinite the charges will not go from one side to the other. They will rather be brought in from infinity.

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  • $\begingroup$ So my diagram is wrong? $\endgroup$
    – Aditya Dev
    Commented Nov 25, 2015 at 14:45
  • $\begingroup$ "Charges are brought from infinity" what will happen to the charges already present at either sides, is net charge conserved here ? $\endgroup$
    – Sujith Sizon
    Commented Nov 25, 2015 at 15:15
  • $\begingroup$ @AdityaDev I think the correctness of the diagram depends on the thinness of the conductior (which is not specified). If it is infinitessimally thin then the is really no upside or downside. Either way there will be spaialy variying charge density, and the diagram does not show that. $\endgroup$
    – Mikael Fremling
    Commented Nov 26, 2015 at 10:12
  • $\begingroup$ @SujithSizon The charges on either sides will rearrange to ensure that the electrostatic potential is zero in the conductor. Since it is infinitely large you may "as much extra charge as you need" with out affecting overal charge conservation. Infinities are strange in this respect. If the plate was big but not infinite, you would presumably note a small charge deficit, at the far edgers of the sample. $\endgroup$
    – Mikael Fremling
    Commented Nov 26, 2015 at 10:15
  • $\begingroup$ I think you can assume the charge to be a an infinite plane have a uniform charge of same magnitude and parallel to the given plate. The charge induced on the plate will be the same. $\endgroup$
    – Aditya Dev
    Commented Nov 26, 2015 at 13:17

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