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This is the question of the classic image problem in which a point charge $q$ is placed at height $h$ above a grounded plate. Only this time the plate is not grounded. When the plate is grounded the force on a charge in the region between the point charge and the plate is proportional to $2q$ because there is equal $-q$ on the sheet. But due to induction an equal $+q$ charge should also be induced on the other side of the plate thereby causing the force to be proportional to only $q$. We can say that this is wrong since the plate is grounded so the induced positive charge will flow down to the earth causing the total force to be proportional to $2q$. But if this plate is not grounded then what shall be the case ? Will the force be proportional to $q$ or $2q$?

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It doesn't matter if the plate is grounded or not, the field outside the plate is the same either way. You know this because the only constraint on that field is that its parallel component at the plate must be zero. That suffices to specify the image-charge field, as long as there is no net charge density on the infinite plate. If there was a net charge density on an ungrounded plate, that just adds a constant field perpendicular to the plate, but if there's no net charge density (i.e., no infinite charge on the plate), whether the plate is grounded or not will simply result in a negligible charge distributed over the whole infinite plate. In short, pulling charge from the ground simply distributes that charge over the whole infinite plate, with negligible effect.

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  • $\begingroup$ But then the force should be proportional to q & not 2q since the effect of induced -q will cancel out the effect of +q. $\endgroup$
    – E2n
    Commented Dec 28, 2016 at 18:14
  • $\begingroup$ The answer is that the field is the same whether the plate is grounded or not, so that seems to be the case you are calling "proportional to 2q." Note that q is also proportional to 2q, so the language is not precise, but I believe I understand what you mean. Note that if you add a net but non-infinite charge to an infinite plate, it has no consequence whatsoever. $\endgroup$
    – Ken G
    Commented Dec 28, 2016 at 18:17
  • $\begingroup$ What I mean is that if the plate is grounded i can say that the induced +q flows to the ground and the negative induced q applies force in the region above the sheet along with the original point charge q ( hence force is proportional to 2q) $\endgroup$
    – E2n
    Commented Dec 28, 2016 at 18:20
  • $\begingroup$ What I mean is that if the plate is grounded i can say that the induced +q flows to the ground and the induced -q applies force in the region above the sheet along with the original point charge q( hence force is proportional to 2q). But if the plate is not grounded then an equal +q is induced on the sheet ( like any normal plate kept in front of a charge ) . The induced charges will give opposite fields in the region on the side of the plate and the point charge and hence their result will be to cancel and force will be only because of the original point charge q(force proportional to q) $\endgroup$
    – E2n
    Commented Dec 28, 2016 at 18:25
  • $\begingroup$ Sure, and if the plate is not grounded, all that happens is the charge on the plate moves around. The net result is all the same fields outside the plate, because either way, the charges in the plate have to cancel the field parallel to the plate. They don't have to cancel the field perpendicular to the plate outside the plate, and inside the plate that is cancelled by the surface charge. The net charge on the plate never matters, it is distributed so as to produce no field anywhere. $\endgroup$
    – Ken G
    Commented Dec 28, 2016 at 18:28

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