Grounding parallel plate conductors

[erisedk]

Homework Statement

So, I always make mistakes on problems such as this (the grounding part), so I'm hoping someone could really explain to me how the process works instead of just answering this particular problem.

There are n large parallel plate conductors carrying charges Q1, Q2,... Qn respectively.
(i) Find the charge induced at surface A (outermost surface, i.e., outer surface of Q1 conductor).
(ii) Find the charge induced at surface B (left surface of second, ie. Q2 conductor)
(iii) If the left conductor (conductor Q1) is earthed, find the magnitude of charge flowing from plate to earth.
(iv) If any conductor is earthed, find the magnitude of charge flowing from plate to earth.

Homework Equations

The Attempt at a Solution

Parts (i) and (ii) are fairly simple. Writing charge distributions on the plates is never an issue for me. But (iii) and (iv) kind of problems trip me up. So, the outermost charge (i.e. the charge on outer surface of plate Q1 i.e. surface A) should flow from the plate to the ground, so the answer should be the same as part (i) but it's not. It's double. So, I'm thinking it's cos some other stuff happens and essentially makes the outermost charge on the rightmost plate (i.e. Qn) also 0. But I'm so confused by it all. Just please someone explain this entire concept to me. And (iv) again I'm pretty confused.

 

[haruspex]

the outermost charge (i.e. the charge on outer surface of plate Q1 i.e. surface A) should flow from the plate to the ground, so the answer should be the same as part (i) but it's not. It's double.

Grounding does not mean the charge will become zero. What will become zero?

 

[erisedk]

The potential

 

[haruspex]

The potential

Right. So given all the other charges, what charge on the grounded plate will produce a zero potential there?

 

[erisedk]

Assuming the distance between the plates to be d, the electric field at the site of the grounded plate just due to the other plates will be (Q2 + Q3 + ... Qn) / 2Aε. So, I want to make the potential zero. What charge, um, I'm not sure? Negative of (Q2 + Q3 + ... Qn) / 2Aε?

 

[haruspex]

Assuming the distance between the plates to be d, the electric field at the site of the grounded plate just due to the other plates will be (Q2 + Q3 + ... Qn) / 2Aε. So, I want to make the potential zero. What charge, um, I'm not sure? Negative of (Q2 + Q3 + ... Qn) / 2Aε?

You're mixing up fields and potentials. You found the field at the first plate generated by the charges on the other plates, but we need to find the potential there due to them.

To be honest, I'm not sure I'm on the right track here. It bothers me that the question does not state the spacing. It also gets tricky because the usual approximation for potential near a large charged plate (a) assumes the plate is uniformly charged - a conducting sheet won't be; (b) only gives the difference between the potential at the plate and the potential at a given distance from it, so it does not give a way to relate to ground potential.

Might need to involve some electrostatics experts.

 

[erisedk]

I wasn't mixing them up. I hesitated to write the potential due to no spacing being given, however, I've encountered some similar problems, and they never usually give the spacing. (a) That's not the intent of the question, just assume it's uniformly charged. (b) Ground potential is zero.
A friend once told me, like a general rule of thumb, that if the outer plate is connected with the earth, the charge on the outermost surfaces will be zero. But I don't really understand it, and besides, I can't do the rest of the work myself either.
Thanks for helping though!

 

[haruspex]

(b) Ground potential is zero.

Yes, I understand that. But if you take the usual infinite uniformly charged plate formula and set the potential to zero at infinity then you get a silly answer. The potential at the plate is infinite.

a general rule of thumb, that if the outer plate is connected with the earth, the charge on the outermost surfaces will be zero.

OK, but that does not mean that the total charge that flows from plate to ground will equal the charge that had been on the outer surface. There may some change to the charge on the inner surface, no?

 

[erisedk]

I'm sorry, I just don't know.

 

[haruspex]

I'm sorry, I just don't know.

Try this: assume it is true that after grounding there will be no charge on the outer surface.
If the charge on that surface had previously been Q, that Q flows to ground.
That change alone would result in a field within the plate.
How much charge would have to flow from the inner surface to ground so that there is no field inside the plate?

 

[erisedk]

Wow! I actually managed to solve the problem.
For (iii) Initial charge on grounded conductor is Q1
Assuming that both outermost plates (ie left surface of Q1 and right surface of Qn) have zero charge after grounding, after writing out the charge distributions for 0 field-
Final charge on grounded conductor is -(Q2+Q3+...Qn)
So, difference is -(Q1+Q2+...Qn) , so +(Q1+...Qn) will flow from earth.
For (iv)
I assumed that the some rth plate (plate Qr) is earthed,
Assuming that both outermost plates (ie left surface of Q1 and right surface of Qn) have zero charge after grounding, after writing out the charge distributions for 0 field-
Initial charge on grounded conductor Qr = Qr
Final charge (on left surface) = -(Q1 + Q2 +... + Qr-1)
Final charge (on right surface) = -(Qr+1 + Qr+2 + ... + Qn-1 + Qn )
Total final charge = - (Q1 + Q2 +... +Qn ) (*Qr missing)
Change in charge = -(Q1 + Q2 + ... + Qn)
So, charge flown = (Q1 + Q2 + ... + Qn)

Numerically, the answers work out. But I don't get:
1. Why do the outermost charges become zero when any conductor, not even necessarily the first, is grounded?
2. I'd appreciate an intuitive explanation why the charge flown from the Earth is independent of which conductor is grounded.

 

[haruspex]

1. Why do the outermost charges become zero when any conductor, not even necessarily the first, is grounded?
2. I'd appreciate an intuitive explanation why the charge flown from the Earth is independent of which conductor is grounded.

The usual approximation for a charge uniformly distributed over a large plate is that the field is constant on one side (reversed on the other) leading to a potential that falls off linearly from the plate (initially). That works well for finding the difference in charges between the two surfaces of each plate: it has to neutralise the field inside the plate.
But it's no good for finding the potentials in the plates here, since we do not know the spacing (and it wouldn't give the desired answer). We have to make the even cruder approximation (for extremely closely spaced plates) that the potential due to a given plate is itself constant, i.e. it generates the same potential in all the other plates as it does in itself.
Now we see that to get a zero potential in a grounded plate it must carry a charge equal and opposite to the sum of the other plates.
If we combine that with our finding for the difference in charges between the two surfaces we find that, for a grounded endplate, all of the charge will be on the inner surface.

 

[erisedk]

OK I understand that for a grounded endplate, all the charge will reside on the inner surface. But I still can't reason why the charge on the outer surfaces of the endplates will be zero even if an internal plate is grounded. I'm sure it's a similar argument, but I'm having a bit of an issue concretely spelling it out.

 

[haruspex]

OK I understand that for a grounded endplate, all the charge will reside on the inner surface. But I still can't reason why the charge on the outer surfaces of the endplates will be zero even if an internal plate is grounded. I'm sure it's a similar argument, but I'm having a bit of an issue concretely spelling it out.

Consider a set-up where one of the plates is grounded. We now add a bit of charge, Q, to an endplate.
The grounded plate will acquire additional charge -Q. That will change the field the grounded plate generates on all the others. The difference in charges of the two surfaces of the endplate will increase by Q to neutralise this field change. Thus all the extra charge that went on the endplate goes to its inner surface.

Another approach: as noted, the approximation used is that all the plates are at the same potential. If anyone plate is grounded then all plates are at zero potential and the sum of their charges is zero.

 

[erisedk]

Thanks! Got it :)