# Ambiguity when taking the Earth as a zero for electric potential

When you ground something in electrostatics, the potential of that body becomes the potential of the Earth once equilibrium has been reached. In this context, it is usually taken that the Earth is at 0V. There are two possibilities for this. Either the constant of integration is chosen such that the Earth is the zero reference point for potential. Or, the potential of the Earth is negligible enough w.r.t. a zero reference at infinity that we take the potentials of the Earth and infinity to be equal: this would imply that the Earth is neutral.

These two options are actually quite different. If we are doing an electrostatics problem with some charged spheres (for instance), we might need to ground one of the spheres: that fixes the potential of that sphere equal to that of the Earth. To calculate the potential of the other spheres in the problem, it is then usually required to do an integral from infinity to the sphere in question. If the Earth were the true (constant of integration) 0 reference, we would instead compute the integral from the Earth to the other sphere instead, however I have never seen this done. In fact, it wouldn't even be feasible since we'd need much more information about the technicalities of the grounding and the surface charge of the Earth.

In that case, what is the nature of the "Earth" in the context of electrostatics? I'm aware of the modelling assumptions that it is a conductor with an infinitely large charge donating/accepting capacity, however do we also take it to always have zero net charge? And do we take the Earth to always be at 0V w.r.t. infinity? Thank you.

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Homework Helper
Gold Member
When I consider a grounded object, I don't worry about the Earth as having some net charge. The Earth and infinity are modeled as reservoirs of charge in the sense that they can accept or donate any (reasonable) number of electrons without a change in their electrostatic potential. Oceans were reservoirs of water drops as they were considered to accept or donate water drops as needed without their level rising significantly but not any more. Anyway, "grounding" an object simply means connecting it to a reservoir of electrons which by definition is at zero electric potential.

When you do a mathematical calculation, you consider your object connected to infinity for zeroing its potential because that makes the integrals easier to handle mathematically. Similarly, when you design a piece of electronics, you do your calculations assuming that zero potential is at infinity. However, when you build the actual gadget, you connect your zero voltage reference to the Earth. Running grounding wires to infinity would be extremely costly, unsightly and ecologically catastrophic. Makes sense, no?

• • nsaspook and etotheipi
So really the word "grounding" within electrostatics is just an abstraction for connecting to an arbitrary reservoir of charges at zero potential w.r.t. infinity. I guess there need not be any mention of the Earth! And the fact that this reservoir is necessarily neutral (otherwise by Gauss' law there would be an electric field, and they would not be at zero potential) means that they do not otherwise influence the chosen system apart from being a source/sink of charge. I suspect this neutrality is preserved since if there are infinitely many charges, removing a few doesn't affect anything!

It follows then the second case is true: the Earth's potential is close enough to zero that when we actually do the experiment it makes no difference, but nonetheless we're not setting our zero reference at the surface of the Earth in our calculations.

That makes sense; it is perfectly valid to set 0V (the constant of integration variety) at the surface of the Earth, but when we need to do integrals with the electric field this becomes unfeasible. For something like circuit theory, however, we don't care about the circuit's potential w.r.t. infinity so it would be fine to set any point in the circuit to the 0V reference!

Thank you!

Staff Emeritus
I hate PF discussions that include the words neutral or ground. We physically tie power circuits to Earth at various points. The reasons have to do with safety and abnormal conditions where something was installed wrong or something failed. Grounding practices sometimes seem more of an art than a science, although the standards bodies are not very artistic. Standard practices also vary widely around the world.

Also as a matter of convenience, we consider Earth as being zero potential.

Massive confusion arises when the two things come together. Some people believe that circuits are grounded because Earth potential is zero.

My preference is to avoid the whole subject.

• etotheipi
Also as a matter of convenience, we consider Earth as being zero potential.

Do you mean w.r.t. infinity? As in an isolated electrically uncharged conductor will have zero potential.

I hate PF discussions that include the words neutral or ground. We physically tie power circuits to Earth at various points. The reasons have to do with safety and abnormal conditions where something was installed wrong or something failed. Grounding practices sometimes seem more of an art than a science, although the standards bodies are not very artistic. Standard practices also vary widely around the world.

Yes, there's so much information online that it's hard to filter out what's important. My reasoning for studying this is more for problem solving in electromagnetism, so I'm only really concerned about the ideal cases and won't delve too far into the practicalities.

Staff Emeritus
Do you mean w.r.t. infinity? As in an isolated electrically uncharged conductor will have zero potential.
There's a third way to get confused by the terms. Yes, we can look at an object like a wire, and count the + and - charges and if they balance declare it neutral. We can't do that with our planet. It is constantly gaining and losing charges from/to space.

I say convenient meaning that we can drop the term in our equations the represents the absolute potential, when it doesn't make any difference to our conclusions. However, there can be cases such as an ionized gas where it is not valid.

• etotheipi
charminglystrange
Can we congratulate eto theipi on becoming an official homework helper, a tremendous accomplishment!
Now he simply needs to replicate this success on maths stack exchange • etotheipi
Now he simply needs to replicate this success on maths stack exchange I have no idea what you're referring to, @charminglystrange...

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• charminglystrange
charminglystrange
• etotheipi