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My question is-

Is the charge on the plates of a parallel plate capacitor induced charge?

Imagine two plates of the capacitor have $+q$ and $-q$ charges on it.

Basically, what I want to know is that if, I earthed (or grounded) one plate of a parallel plate capacitor, would the charge distribution still remain symmetrical (in equal and opposite sense) or would it change? I think that if it is induced charge it would not change otherwise it should change.

Thanks!

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Imagine two plates of the capacitor have $q$ and $-q$ charges on it.

If both plates have some non-zero net charge, it means that the charge is not induced. Electrostatic induction could cause charge redistribution in a neutral conductor, but the net charge would remain zero.

Basically, what I want to know is that if, I earthed (or grounded) one plate of a parallel plate capacitor, would the charge distribution still remain symmetrical (in equal and opposite sense) or would it change?

Grounding of one of the plates would not change charge distribution. Moreover, if the grounded plate did not have any net charge on it before the grounding, it would attract charge $q$ (positive or negative, depending on the charge sign on the other plate) from the ground to make the capacitor neutral and the distribution of charges would be indistinguishable from the first scenario.

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Basically, what I want to know is that if, I earthed (or grounded) one plate of a parallel plate capacitor, would the charge distribution still remain symmetrical (in equal and opposite sense) or would it change?

The energy associated with an unbalanced charge tends to be relatively high. Look at all of the matter around you. It consists of positively charged protons and negatively charged electrons but virtually all of the objects around you are electrically neutral because of the large energy penalty associated with having unbalanced charges. So, yes, if you started charging one plate of a parallel plate capacitor with a static electricity generator with the opposite plate connected to ground, then the opposite plate would try to draw in electrons from the ground or expel electrons into the ground in order for the capacitor as a whole to remain electrically neutral and thereby minimize the electrostatic energy of the system. Your attempt to charge up one plate of the capacitor would induce a nearly equal and opposite charge on the opposite plate of the capacitor.

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