Why do we assume that Earth and infinity are at same potential?

In many standard texts I have seen the approach of taking any point connected with Earth in a circuit as 0 potential. Obviously, there is nothing special with 0 as long as we are concerned with potentials except that it is standard potential for infinity when there are only finite charges. I have seen a lot of question assigning same potential to both the Earth and infinity: such as this one 'A conducting sphere is initially charged with charge q. The sphere is then placed in a uniformly charged hollow sphere having charge q2 . The solid sphere in the interior is then connected to Earth with a conducting wire.There are no other charge distributions. Find the final charge on the sphere.'

• Have you seen both Earth and infinity be set to $0\ \rm{V}$ in the same example without there being a connection between them? – md2perpe Nov 6 '18 at 16:51
• @md2perpe yes i have seen a lot of such questions. Seems like they were wrong – N.S.JOHN Nov 7 '18 at 3:42
• @md2peprpe what is the potential of Earth when we take that at I finity is zero? – N.S.JOHN Nov 10 at 5:07
• Unfortunately I don't know. – md2perpe Nov 10 at 8:14

We can take any point of our choice to be the zero potential. When we are working with circuits down on Earth we usually take the ground on Earth as the zero potential. When we are doing electrostatics or electrodynamics we usually take infinity to have the zero potential. It's just a matter of choice for the specific situation.

• I suggest that you read my question. I am asking why we take both the infinity and Earth tho be at 0 potential. There are many questions using this approach – N.S.JOHN Nov 6 '18 at 16:44
• @N.S.JOHN. I did read your post. There is no question why we take both infinity and Earth to have the zero potential. The only question in your post asks whether we mark Earth as $0\ \mathrm{V}$ just for reference. – md2perpe Nov 6 '18 at 16:50
• Then try to answer this question: 'A conducting sphere is initially charged with charge $q$. The sphere is then placed in a uniformly charged hollow sphere having charge $q_{2}$ . The solid sphere in the interior is then connected to Earth with a conducting wire.There are no other charge distributions. Find the final charge on the sphere.' Check if you are not assuming infinity and Earth to be at the same potential while solving this question. – N.S.JOHN Jan 6 at 11:08
• @N.S.JOHN. I'm not that good at doing electrostatic calculations. Do you have a calculation that I can study to find if there really is a need to assume that infinity and Earth have the same potential or if there is some reason for the assumption? – md2perpe Jan 12 at 8:11
• The usual solution to the above question is to find the charge on the solid sphere so as to make the potential on its surface( and interior 0). Now can you see it?" – N.S.JOHN Jan 12 at 13:36

Why do we assume that the Earth and Infinity are at the same potential?

However, for different problems we do assign zero potential to infinite distance from objects of interest and to Earth (or anything else convenient).

For example, in computing the escape velocity of this earth this is typically done. In other sorts of problems involving Gravity, for example, Earth's surface is taken as 0 potential so that simple equations such as potential energy can be represented by: $$m g h$$.

With problems in electricity, we assign zero potential to any convenient location for the problem. In a circuit, such as used in a cell phone, the zero potential is nothing more than a common conductor considered as "ground" for the circuit. However, if I were to have another device, say it is wired to a grounding rod, and I call that zero potential, I could easily (sometimes) measure a potential difference between these two zero potentials (i.e. cell phone and grounding rod). This is because there are other electrical (e.g. static charges on phone) effects involved however both points are valid zero potentials for the problems they are solving.

Obviously, we all know that it is only the potential difference that is realistically involved in these problems. We can add to the absolute potential any finite value we want, whether positive or negative or zero, and the physics that we measure, observe, and work with does not change.

It is not clear whether you are asking about assigning zero potential to infinity and earth simultaneously or not. However, you can do that and even use those definitions in the same problem arena. Although you are assigning zero potential to infinity and Earth simultaneously you are not assigning it to the same type of potential.

• Then try to answer this question: 'A conducting sphere is initially charged with charge q. The sphere is then placed in a uniformly charged hollow sphere having charge q2 . The solid sphere in the interior is then connected to Earth with a conducting wire.There are no other charge distributions. Find the final charge on the sphere.' Check if you are not assuming infinity and Earth to be at the same potential while solving this question – N.S.JOHN Jan 6 at 15:22

A simple example with equipotential lines, which include infinity by construction:

The electric potential of a dipole show mirror symmetry about the center point of the dipole. They are everywhere perpendicular to the electric field lines.

The straight line is zero potential by the addition of the two charges, +1/r and -1/r, and is common all the way to infinity.

The problem at hand decides the zero of the potential, and there is no contradiction to having one at infinity as this simple example shows by the infinite zero potential points on the line.

This illustrates that it is the boundary conditions of the specific problem that decide where the zeros of the potential are, and there is no problem with having a zero at infinity and zeros at close by.points.

• what does this have to do with my questiuon? – N.S.JOHN Jan 7 at 7:02
• That it is the boundary conditions of the specific problem that decide where the zero potentials are, and there is no problem with a zero at infinity and a zero at close by.points. I am not going to solve your added problem, if the problem demands a zero at infinity, a zero it is. – anna v Jan 7 at 7:04

You can consider the earth to be at zero potential because it can be thought of as an infinite source and sink.

This means that if you apply positive volts, then the current will flow into the ground and if you apply negative volts, current will flow from the ground.

You should also know that the potential of a sphere drops at $$1/r$$ outside the sphere. Since this is the only charge in the universe therefore the potential at infinity is also zero.