Understanding Earthing: Impact on Charged Bodies & Charge Distribution

[person_random_normal]

I understand earthing as - if we Earth a charged body, its charge distribution changes such that, its potential becomes zero.

So now I have a doubt that if we have 2 charged spheres(Q1 , Q2) ,separated by some distance and then if we Earth one of them , then how will the charge distribution change ?
On similar lines, what will happen when 3 charged spheres kept in the same way and then if I Earth
1 the middle one
2 the one at extreme
??

 

[haruspex]

Are you asking what the charge distribution looks like on two conducting spheres near each other? I don't think there's a closed form solution for that. You can get successively better approximations using the method of images.
Or are you simply asking how earthing one affects it? Before earthing, each is at a uniform potential. After earthing, each is at a uniform potential. So the charge distributions are essentially similar. It might be possible to figure out the charge ratio (as a function of the radii and the distance beteween the spheres) to get one of the potentials to be zero.

 

[person_random_normal]

using the method of images.

What is method of images ??

 

[haruspex]

What is method of images ??

Lots of references on the net. See e.g. http://farside.ph.utexas.edu/teaching/em/lectures/node64.html. They explain it just as well as I could.

 

[person_random_normal]

I don't think there's a closed form solution for that

?
there will be only one solution this !

I agree that there are many possible charge distributions allowing potential of the earthed body zero , but only one of that is naturally feasible , obeying Uniqueness theorem ! (this is all i know about uniqueness theorem)

So what I exactly want to know is - which will be the right charge distribution for the situation I mentioned ??

 

[haruspex]

?
there will be only one solution this !

I agree that there are many possible charge distributions allowing potential of the earthed body zero , but only one of that is naturally feasible , obeying Uniqueness theorem ! (this is all i know about uniqueness theorem)

So what I exactly want to know is - which will be the right charge distribution for the situation I mentioned ??

I didn't say there would not be a unique solution, I said it would not be expressible in 'closed form'. https://en.m.wikipedia.org/wiki/Closed-form_expression

 

[person_random_normal]

Ok

But now have you understood what exactly my doubt is ?

 

[haruspex]

Ok

But now have you understood what exactly my doubt is ?

I'm not sure what sort of answer you are looking for to the original question. Do you want to find a specific equation for the distribution, or just a verbal description of how earthing one will change it?

Does this work: Suppose that before earthing the charges are -Q and +Q, and the potentials -V and +V. Zero potential is just a convention, so if after earthing the -Q one it is at ground and the other at V', just reset the 'zero' to V'/2. Then you have potentials -V'/2 and +V'/2. That's the same as you would have had if you had started with charges -+QV'/2V. Then add back equal uniformly distributed charge to each to bring the unearthed one back to +Q. Does that make sense?

 

[person_random_normal]

Check out the attached file,

there , after grounding 3 work done to get a test charge from infinity to 3 will be zero, but in for moving the test charge -from infinity to 1 ; from 1 to 2 ; from 2 to 3 , if work done is W1 , W2 , W3 respectively , the what will be relation between W 1, W2, W3
OR
Will W1=W2=W3= 0 ??

I want to know the general idea of nature for deciding charge distribution in such kind of systems !

 

[haruspex]

Check out the attached file,

there , after grounding 3 work done to get a test charge from infinity to 3 will be zero, but in for moving the test charge -from infinity to 1 ; from 1 to 2 ; from 2 to 3 , if work done is W1 , W2 , W3 respectively , the what will be relation between W 1, W2, W3
OR
Will W1=W2=W3= 0 ??

I want to know the general idea of nature for deciding charge distribution in such kind of systems !

Try this: http://rspa.royalsocietypublishing.org/content/468/2145/2829

 

[person_random_normal]

Can you put that qualitatively !

 

[ehild]

Check out the attached file,

there , after grounding 3 work done to get a test charge from infinity to 3 will be zero, but in for moving the test charge -from infinity to 1 ; from 1 to 2 ; from 2 to 3 , if work done is W1 , W2 , W3 respectively , the what will be relation between W 1, W2, W3
OR
Will W1=W2=W3= 0 ??

The charge distributions on the spheres will change if you ground one of them, but the charge remains the same on the isolated spheres, and all points of a sphere are at the same potential. If these potentials are V1, V2 and V3=0, the work done when you bring in a test charge from infinity to sphere 1 is V1=W1.
When moving the test charge from infinity to sphere 2 the work is V2, so the work when moving the test charge from sphere 1 to sphere 2, is W2=V2-V1.
Sphere 3 is at zero potential so the same work is done when the test charge is moved from sphere 2 to sphere 3, as from sphere 2 to infinity, that is -V2. Your W3=-V2.
The total work between infinity and the grounded sphere is zero, but the individual works are not..

 

[person_random_normal]

Ok
But now if we have n infinitely large plates placed parallel to each other at equal distances, with charges Q1, Q2. . . . . . .
And then if we ground r th plate then what will be subsequent charge distribution? ?

 

[ehild]

The charge distribution on the infinite parallel plates has to be uniform. In case the charges are finite, the surface charge densities are zero.

 

[person_random_normal]

What will be the charge on the grounded plate ?

 

[haruspex]

The charge distributions on the spheres will change if you ground one of them, but the charge remains the same on the isolated spheres, and all points of a sphere are at the same potential. If these potentials are V1, V2 and V3=0, the work done when you bring in a test charge from infinity to sphere 1 is V1=W1.
When moving the test charge from infinity to sphere 2 the work is V2, so the work when moving the test charge from sphere 1 to sphere 2, is W2=V2-V1.
Sphere 3 is at zero potential so the same work is done when the test charge is moved from sphere 2 to sphere 3, as from sphere 2 to infinity, that is -V2. Your W3=-V2.
The total work between infinity and the grounded sphere is zero, but the individual works are not..

Perhaps I misunderstood (Shreyas had not made the question precise) but I thought the problem here was to find the potentials given the charges, radii and separations. Or, failing that, to make some qualitative statement about how the potentials change when one of the spheres is grounded. I see from post #11 that such a qualitative description may be what is sought, but even that is not straightforward.
If the grounded sphere had been at a positive potential then clearly it will go to a lower potential. Seems also clear that the potentials will become lower at the other spheres. But will they drop by as much?

 

[haruspex]

What will be the charge on the grounded plate ?

Infinite parallel plates are much easier. What is the potential at distance x from a single charged infinite plate?

 

[ehild]

Infinite parallel plates are much easier. What is the potential at distance x from a single charged infinite plate?

Potential with respect to what? What would be the potential of a single charged infinite plate at infinity?
Instead of infinite plates we should say that the extension of the plates are very large with respect to the distance between them. In that case is it possible that the plates are at non-zero potentials with respect to infinity.

 

[haruspex]

Potential with respect to what?

For infinite uniformly charged plates, it is usually taken as relative to the potential at the plate. The potential at infinity becomes +/-infinity.

 

[ehild]

For infinite uniformly charged plates, it is usually taken as relative to the potential at the plate. The potential at infinity becomes +/-infinity.

What is the situation if the infinite plate gets finite charge q?

 

[haruspex]

What is the situation if the infinite plate gets finite charge q?

It doesn't mean anything to say a finite but nonzero charge is uniformly spread on an infinite surface. It's like trying to pick an integer at random, all equally likely.

 

[ehild]

Well, a finite charge uniformly distributed over an infinite area results in zero surface charge density. Don't you agree?
In Classical Electrodynamics, charge is not quantised.

All points of a conductive plate are at the same potential. If the plate is infinite, its potential is the same at infinity as it is at a given point. P, as you do not need to do work when moving a test charge along the conductive plate.

 

[person_random_normal]

What made me start the thread was - that problem of n infinitely large plates placed parallel to each other at equal distances.

So now let me be very clear!
Before I tell you what problem I exactly have with that infinitely large plates problem, let me tell you one thing that I observed for infinitely large plates -
If we have 2 infinitely large plates(they are truly infinite no approximation) placed parallel to each other, and if we ground one of them, then potential of only one plate can become zero ie we approach the grounded plate from two sides say we are approaching from right(where we encounter the grounded plate directly when we approach from infinity, not the non grounded plate first and then the grounded one) so then if the potential of the grounded plate is to be zero then electric field to the right of the grounded plate has to be zero. But charge distribution favouring this won't make work done zero if we approach the grounded plate from left.
I hope I'm clear !

 

[person_random_normal]

So now my doubt
As of the problem where n infinitely large plates placed parallel to each other at equal distances
There if we choose some r th plate having charge Q r and ground it then what will be subsequent charge distribution?

Answer given is Σ Q i =(Q1 + ...+Qn)
This much charge will go into ground , from that r th plate.

Why so ?

 

[DrZoidberg]

.. But charge distribution favouring this won't make work done zero if we approach the grounded plate from left.

Of course the work is not zero. If we try to add charge to the left plate (the one that is not grounded if I understood you correctly) and we come from infinity and approach from the left, then there is some work required to do this. The larger the plates are, the smaller the field strength on the left side becomes but at the same time the field strength drops off more slowly so assuming a constant charge density on the plate the work required will stay the same. If the size of the plate approaches infinity the field strength will approach zero but to calculate the required energy per charge (i.e. the voltage) you have to integrate that field strength over an infinite distance. So you end up with infinity times zero which (in this case) is not equal to zero.

 

[person_random_normal]

Of course the work is not zero. If we try to add charge to the left plate (the one that is not grounded if I understood you correctly) and we come from infinity and approach from the left, then there is some work required to do this. The larger the plates are, the smaller the field strength on the left side becomes but at the same time the field strength drops off more slowly so assuming a constant charge density on the plate the work required will stay the same. If the size of the plate approaches infinity the field strength will approach zero but to calculate the required energy per charge (i.e. the voltage) you have to integrate that field strength over an infinite distance. So you end up with infinity times zero which (in this case) is not equal to zero.

I am fine with it !

 

[person_random_normal]

But I want you to consider my doubt in light of the that one sided phenomenon which we have discovered.

 

[DrZoidberg]

I wasn't aware we discovered any phenomenon in this thread.
Why don't you just use an equivalent circuit diagram to analyze your examples? Each plate has a certain capacity relative to infinity and a capacity relative to it's adjacent plates. For an infinite plate that's surrounded by two other plates the capacity relative to infinity is zero. With the capacities and charges you can calculate all the voltages. And once you have that you can determine the field strengths and from that how much charge is on each of the two sides of any plate.

 

[person_random_normal]

Have you got my point?

 

[ehild]

Have you got my point?

If one plate is at zero potential and the other one at V potential with respect to it, you need Vq work to move a charge q from the grounded plate to the non-grounded one.
But the potential is zero also at infinity, so the same work is needed to bring in the charge q from infinity to the non-grounded plate along its 'left side'.
You can not assume infinite plate. If you cut the universe into two, that plate still would be finite.
Assume finite plates instead, with lateral sizes much larger than the separation between them. In this case, you can assume finite charge densities, homogeneous far from the edges.
Assume two parallel plates of area A. Neither plates are grounded. The charge on the left plate is Q1 and the charge per unit area is σ1=Q1/A. Near the plates, the electric field due to the left plate would be -σ1/(2ε₀) on the left and σ1/(2ε₀) on the right (the blue arrows). The other plate has charge Q2, charge per unit area σ2=Q2/A, and electric field due to it is -σ2/(2ε₀) on the left and σ2/(2ε₀) on the right of it (the red arrows).
The resultant field is the sum of the contributions from both plates. On the left, it is E1=-(σ1+σ2)/(2ε₀). On the right, it is E2=(σ1+σ2)/(2ε₀).
Between the plates, the electric field is E=(σ1-σ2)/(2ε₀). The potential difference between the plates is V=(σ1-σ2)/(2ε₀)d.
Far away, the arrangement looks as a tiny charged object, of net charge Q1+Q2, and the electric field tends to zero as the distance increases.

Assume the right plate is grounded. Then E2=0, so the charge per unit area should be -σ1 on it. The electric field between the plates becomes E=σ1/ε₀, and the potential difference between the plates is V=σ1/ε₀d. You see, that the charge from the ground makes the net charge zero. Also E1=0.
You can extend this method to any number of parallel plates close to each other. The charge moving onto the grounded plate makes the whole arrangement neutral, the net charge of the whole arrangement becomes zero.