Point-like charge between grounded plates: Induced charges

[Jacob White]

Homework Statement

I have came across such a method:
Suppose we have two connected and grounded very large plates(like in capacitor). And we put a point like charge in the plane parallel to the plates and at distance x from one of them. The question is what would be the charge induced on each of the plates?
They state that the charges induced on each plate would no be altered by moving a charge in a initial plane and their explain it only by one word - superposition. It is useful because it means that the induced charges would be the same as the intial charge was spread out uniformly in a plane and equivalently we can solve a much simpler problem.

Relevant Equations

Superposition

I don't understand why moving charge doesn't affect the magnitude of induced charge. Could someone explain why it is possible? Thanks in advance.

 

[Jacob White]

There is even another thread on PF on this problem where they suggested method of images however it is much more complicated and it is definitely not they had in mind.

 

[haruspex]

I don't understand why moving charge doesn't affect the magnitude of induced charge. Could someone explain why it is possible?

I'm not certain what you are asking. Your question as quoted above does not seem connected with superposition; that is to do with adding up the induced distributions due to each point charge.
Clearly, if you have a single point charge then the induced charge density at some patch of a plate depends on where the patch is in relation to the point, so if you move the point charge around, keeping it a constant distance from the plates, then the induced charges move with it.
If you move the point charge towards one plate then that is different, but you can use a Gaussian surface to show the total induced charge is equal and opposite to the point charge.

 

[Jacob White]

I don't know why moving the point charge around, keeping it a constant distance from the plates keeps the propotion of induced charges at plates. It looks like it would be that way but I wonder if we can prove that somehow.

 

[TSny]

They state that the charges induced on each plate would no be altered by moving a charge in a initial plane and their explain it only by one word - superposition.

I don't understand why moving charge doesn't affect the magnitude of induced charge. Could someone explain why it is possible? Thanks in advance.

I think they are assuming the plates are large enough that you can neglect "edge effects". If you were to slide the point charge on the plane so that it is close to the edge, then I think the net induced charge on each plate would change (compared to having the point charge far away from the edges). But as long as you keep the point charge away from the edges, then the net induced charge on each plate will remain essentially constant as you slide the charge around on the plane. If the plates were infinite plates (no edges), it should be clear that the net induced charge on each plate is independent of where the point charge is placed on the plane between the plates.

The principle of superposition that they are using is that if you place two point charges between the plates, then the net charge on the upper or lower plate is the sum of the net charges that each charge alone would induce on the plate. This does not require the point charges to be on the same plane or far away from the edges.

 

[Jacob White]

Yes, in the case of infinite plates definitely moving charge will not affect induced charges. However originally they stated that the area of plate is A so it's finite.

 

[TSny]

Yes, in the case of infinite plates definitely moving charge will not affect induced charges. However originally they stated that the area of plate is A so it's finite.

Yes. However, they stated that the plates are "very large". This is often interpreted as meaning "neglect edge effects".

 

[Jacob White]

Ok, but if didn't assume that they are very large what would happen? Do you have any idea how we could deal with situation like this?

 

[TSny]

Ok, but if didn't assume that they are very large what would happen? Do you have any idea how we could deal with situation like this?

I think it would be too complicated to deal with in a simple way. Any time we take the electric field of a parallel-plate capacitor to be uniform everywhere between the plates, we are neglecting edge effects. If we can't neglect edge effects, then equations such as $E_1/E_2 = (d-x)/x$ are not going to be applicable.

 

[Jacob White]

Ok, thank you for explanation!