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

• etotheipi
In summary, the concept of grounding in electrostatics involves connecting an object to a reservoir of electrons at zero potential, which is often assumed to be the Earth. However, this is mainly for mathematical convenience and does not necessarily reflect the actual potential of the Earth. The Earth is modeled as a neutral charge reservoir in electrostatics, and its potential is usually considered negligible in relation to infinity. Confusion can arise when the practicalities of grounding are taken into account, but for the purposes of problem solving in electromagnetism, the ideal cases are usually studied.
etotheipi
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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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!

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
anorlunda said:
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.

anorlunda said:
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.

etotheipi said:
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
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
charminglystrange said:
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
etotheipi

## 1. What is meant by "zero for electric potential" when referencing the Earth?

When we say "zero for electric potential" in relation to the Earth, we are referring to the fact that the Earth is often used as a reference point for measuring electric potential. This means that the Earth is considered to have a potential of zero, and all other points are measured relative to this point.

## 2. How does using the Earth as a zero for electric potential affect calculations?

Using the Earth as a zero for electric potential simplifies calculations and allows for easier comparison between different points in an electric field. It also eliminates the need for a separate reference point, as the Earth is always present and accessible.

## 3. Can the Earth's potential ever change?

Yes, the Earth's potential can change due to various factors such as lightning strikes, changes in the Earth's magnetic field, and human activities such as power generation and distribution. However, for most practical purposes, the Earth's potential can be considered constant.

## 4. Is the Earth the only reference point for electric potential?

No, the Earth is not the only reference point for electric potential. In some cases, other reference points may be used, such as the center of an electric dipole or the surface of a charged conductor. However, the Earth is commonly used as a reference point due to its widespread availability and relatively stable potential.

## 5. What are the implications of using the Earth as a zero for electric potential in practical applications?

Using the Earth as a zero for electric potential allows for easier measurement and comparison of electric potential at different points. It also simplifies the calculation of electric fields and allows for the use of simpler equations. However, it is important to note that the Earth's potential may not be uniform in all locations, so adjustments may need to be made in certain situations.

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