# Potential of a grounded conductor in the presence of an external charge

• kmm
In summary: That is correct if we assume an ideal ground, that is a ground that can hold its potential to zero, regardless of external charges. This means that the ideal ground is a perfect conductor that has infinite dimensions.In summary, when a conductor is grounded, the potential is still V=0, even in the presence of an external charge. However, the potential changes depending on the magnitude of the external charge.

#### kmm

If we set the potential at infinity to be zero, we find that the potential of a grounded conductor is V=0. The conductor being grounded has no net charge and produces no external field, so I understand why in that situation we would say the potential of the conductor is zero.

However, in studying the classic image problem, we take a charge above an infinite, grounded conductor and assume the conductor to be at V=0 because it's grounded. In all the explanations I've seen of this, there's no explanation of why we can still take the potential to be zero, but that this is 'known' or 'obvious'.

So if we have a conductor that is grounded, but in the presence of a charge, we get some induced charge on the conductor and it's not obvious to me that the potential remains at zero here, in general. In the image problem it's shown that the image charge configuration does produce a zero potential at the conductor, but we got this by initially assuming the boundary condition that the potential would be zero at the conductor.

The only explanation I can see for why we can always assume that the grounded conductor has an electric potential of zero is that a conductor is an equipotential. Therefore if we tie a conductor to ground, where an effectively infinite amount of charge can be dissipated and is at zero potential, the entire conductor must remain at zero potential since the potential must be the same everywhere. Am I thinking about this correctly?

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Delta2
If there is a potential difference between two locations with a low-resistance connection between them, what happens?

kmm
berkeman said:
If there is a potential difference between two locations with a low-resistance connection between them, what happens?
In that case there is current in the conductor. So even though charge on a grounded conductor has rearranged itself under the influence of an external charge, the potential must still be the same as ground (V=0) since there is no current. Therefore anywhere on the conductor, regardless of the charge configuration, must be equal to ground potential, or zero. Well, now it seems obvious..

Though the ground and the conductor will always be equipotential, this doesn't necessarily means that their potential will always be zero and constant , regardless of the external charge. I believe their potential will change depending on the magnitude of the external charge.

What will happen if we bring infinite external charge?

Delta2 said:
Though the ground and the conductor will always be equipotential, this doesn't necessarily means that their potential will always be zero and constant , regardless of the external charge. I believe their potential will change depending on the magnitude of the external charge.

When we ground a conductor, aren't we saying we are dissipating all charge? In that case, what is producing a nonzero potential?

Delta2 said:
What will happen if we bring infinite external charge?

Wouldn't bringing in infinite external charge be effectively the same as maintaining current in the conductor?

kmm said:
When we ground a conductor, aren't we saying we are dissipating all charge? In that case, what is producing a nonzero potential?
The external charge is producing a nonzero potential but it is producing small potential if the charge is small. That was my point.

Wouldn't bringing in infinite external charge be effectively the same as maintaining current in the conductor?
I am not sure for this and can't comment on this.

Delta2 said:
The external charge is producing a nonzero potential but it is producing small potential if the charge is small. That was my point.

OK, when you said regardless of external charge, I thought you meant that we could have nonzero potential in a grounded conductor in the absence of external charge. But again since the conductor is grounded, even in the presence of external charge, the conductor should still have a potential of zero since ground is at zero. This is also the assumption made in the classic image problem where the potential at the conductor must be zero for some charge q above it, because the conductor is grounded. Is that not right?

kmm said:
OK, when you said regardless of external charge, I thought you meant that we could have nonzero potential in a grounded conductor in the absence of external charge. But again since the conductor is grounded, even in the presence of external charge, the conductor should still have a potential of zero since ground is at zero. This is also the assumption made in the classic image problem where the potential at the conductor must be zero for some charge q above it, because the conductor is grounded. Is that not right?
That is correct if we assume an ideal ground, that is a ground that can hold its potential to zero, regardless of external charges. This means that the ideal ground is a perfect conductor that has infinite dimensions.

But in reality there are no ideal grounds, even Earth has finite dimensions and it's not a perfect conductor.

kmm
Delta2 said:
That is correct if we assume an ideal ground, that is a ground that can hold its potential to zero, regardless of external charges. This means that the ideal ground is a perfect conductor that has infinite dimensions.

But in reality there are no ideal grounds, even Earth has finite dimensions and it's not a perfect conductor.
Definitely. In this case, I was assuming an ideal situation.

Delta2
kmm said:
When we ground a conductor, aren't we saying we are dissipating all charge?
No. The grounded conductor can still have an induced charge. Do a google search for electrophorus, for example. The justification that the potential is uniform throughout the conductor is based on the notion that the net force exerted on free charges within the conductor is zero. Choosing a value of zero for this potential is a convention, it's not physical.

Mister T said:
No. The grounded conductor can still have an induced charge. Do a google search for electrophorus, for example. The justification that the potential is uniform throughout the conductor is based on the notion that the net force exerted on free charges within the conductor is zero. Choosing a value of zero for this potential is a convention, it's not physical.
Yes the electrophorus is real and illustrative, but I fear it will confuse the OP who I think was asking about the metal touching ground directly without the dielectric plate in the middle.

@MMM does the electrophorus confuse you?

Mister T said:
No. The grounded conductor can still have an induced charge. Do a google search for electrophorus, for example.

Right, I meant net charge.

Mister T said:
The justification that the potential is uniform throughout the conductor is based on the notion that the net force exerted on free charges within the conductor is zero. Choosing a value of zero for this potential is a convention, it's not physical.

Yes, I was considering the condition where our reference point was at infinity that we set to zero.

anorlunda said:
Yes the electrophorus is real and illustrative, but I fear it will confuse the OP who I think was asking about the metal touching ground directly without the dielectric plate in the middle.

@MMM does the electrophorus confuse you?
No it doesn't, and at this point my initial confusion has been cleared up.

kmm said:
Right, I meant net charge.
So did I. A grounded conductor can have a nonzero net charge. Say you have a conducting sphere and you ground it. Then rub a balloon on your hair and bring it near (without touching) the sphere. It will induce a net charge on the sphere that is opposite in sign to the charge on the balloon.

Mister T said:
So did I. A grounded conductor can have a nonzero net charge. Say you have a conducting sphere and you ground it. Then rub a balloon on your hair and bring it near (without touching) the sphere. It will induce a net charge on the sphere that is opposite in sign to the charge on the balloon.
Sorry for the confusion here. Yes, I understand that. When I initially asked the question about a grounded conductor dissipating all charge, I had misunderstood Delta2 in post #4, thinking that they were implying we could have net charge on a grounded conductor even in the absence of some external charge or field. I was incorrectly responding to you in that context.

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