Why don't induced charges cancel each other out?

[OmegaKV]

If you bring a positive charge near a thin conducting sheet, it will induce a negative charge on the side nearest to the charge. And in order for the conducting sheet to remain electrically neutral, a positive charge of equal magnitude will be induced on the other side.

How could the conductor have any significant effect at all on the total electric field? It seems to me like the field from the induced positive and negative charges should "cancel out" everywhere except for points very close to the conducting sheet (i.e. points not much greater than the thickness of the sheet).

 

[vanhees71]

Have a look at the multipole expansion of the electrostatic field. It starts with a monopole contribution, i.e., a Coulomb field with the total charge of the body as charge in the Coulomb field. Then comes a dipole term, which you can depict as two opposite point charges very close to each other, then a quadrupole term which you can depict as two dipoles very close to each other and so on. Except the very first monopole term the sources of all other terms are equivalent to charge distributions with total charge 0.

There are two examples, where you can solve this kind of problem exactly: A point charge in front of a infinite conducting half-plane or a point charge close to a conducting sphere. Having solved the problem for a point charge, you have in principle also solved it for any charge distribution close to the conductor since what you evaluate there is the Green's function of the electrostatic problem with given boundary conditions due to the conductor.

 

[Dragon27]

If you bring a positive charge near a thin conducting sheet, it will induce a negative charge on the side nearest to the charge. And in order for the conducting sheet to remain electrically neutral, a positive charge of equal magnitude will be induced on the other side.

Why are you sure that positive and negative charges will have equal distribution on the whole sheet? Intuitively, we may expect more negative net charge at the point below the positive charge, and more positive net charge towards the edges of the sheet.