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In the classic image charge problem of a charge $q$ above an infinite grounded plane, it is well known that the field lines essentially behave as though there were a negative charge behind it, and the total charge on the plate is $-q$.

enter image description here

However, since the plate is grounded wouldn't the extra negative charge go back to the earth? How can the plate remain negatively charged even when connected to ground?

Further, how would the situation change if the plate were not grounded? I assume the positive charge would induce a negative charge on one side the plane, but the total charge would still be zero. Would the field lines terminate in this case?

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    $\begingroup$ The negative charge on the grounded plate came from the earth. The act of connecting the plate to ground is what allowed the negative charge to relocate to the plate's surface by the attraction of the positive charge above it. Grounding doesn't equate with "there is zero charge." $\endgroup$
    – user55515
    Commented Apr 3, 2016 at 23:26
  • $\begingroup$ @user55515 Ah that makes sense. And then am I correct in assuming that if the plane were not grounded, it would induce a negative charge on one side and a positive charge on the other side, and while the field inside the conductor would be zero it would re-appear on the other side? $\endgroup$
    – 1110101001
    Commented Apr 4, 2016 at 5:50
  • $\begingroup$ Indeed. That sounds correct to me. $\endgroup$
    – user55515
    Commented Apr 6, 2016 at 3:34
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    $\begingroup$ What if the plate is not grounded? Will we still be able to use method of image charge? $\endgroup$ Commented May 5, 2016 at 5:08

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You have a small misconception. Grounding the metal plate does not mean that the plate will always be neutral. It just means that the potential of the plate will be equal to that of the earth.

When you place a charge nearby, the potential of the metal plate changes from that of the earth. This creates a potential difference between the metal plate and the earth due to which some charge is transferred from the earth to the metal plate so that the potential difference becomes zero.

If the plate was not grounded, the net charge of the plate would be zero. But since it is a metal plate, the potential throughout the plate should be constant. Due to this condition, a charge redistribution would take place in the presence of the external charge so that the metal plate is at a constant potential. But note that the metal plate need not have the same potential as before.

Griffith's has a good mathematical introduction to such problems.

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