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 $-q$.
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?
Best Answer
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.