# Definition of electrostatic potential difference

Everyone knows that the potential difference between two points 1 and 2 is given by $$\int_{1}^{2}{\vec{E}}\cdot d\vec{l}$$. My question is that can we calculate the potential difference between two point charges by this formula? Say, one of them is +ve charge and the other is a -ve charge.

Surely the end points give divergence. However, excluding those end points, everything is fine between those two points. Can the potential be defined in some limit?

-Neel

## Answers and Replies

Everyone knows that the potential difference between two points 1 and 2 is given by $$\int_{1}^{2}{\vec{E}}\cdot d\vec{l}$$. My question is that can we calculate the potential difference between two point charges by this formula? Say, one of them is +ve charge and the other is a -ve charge.

Surely the end points give divergence. However, excluding those end points, everything is fine between those two points. Can the potential be defined in some limit?

-Neel

Your case sounds like the electric dipole where the constant potential can be ploted as closed contours around the two charge where:

$$V \;=\; \frac{\vec p \cdot \vec R }{ 4 \pi \epsilon_0 R^2}$$

If the charge is not the same, you can still use the way they derived the dipole potential and come up with the formula. Or if you don't assume the distance of the point P where you measure the potential is much bigger than the distance d between the two charges, you just don't make the assumption like the dipole formula. Things get more complicated, but I think that would be the way to start.

BTW $$V_{21} = -\int_{1}^{2}{\vec{E}}\cdot d\vec{l}$$.

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Probably I did not mean that...my point is the difference in potential between those two charges...

Probably I did not mean that...my point is the difference in potential between those two charges...

You mean the potential difference between a +ve charge and a -ve charge, not necessary existing together?" I am unease on this question because if you really mean potential at the point, you cannot do that because and the point charge where the distance become zero, the function is not defined. Even your formula is meant for potential at some point P some distance away from the charge.

I think you should make a drawing or explain a little bit more.

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Yes I know there is a big problem with the divergence at the end points. There are thousands of paths between a source charge and a sink charge, right? Let there is no other charge in the scene. I was thinking that these paths are all equivalent in the sense that the potential difference along all these paths should be equal to each other. This can give rise to a variational principle. However, probably that is not the case; along each different path, the potential difference is infinite. But we cannot say infinite=infinite. May be a point close enough to the charge will satisfy the condition.

Yes I know there is a big problem with the divergence at the end points. There are thousands of paths between a source charge and a sink charge, right? Let there is no other charge in the scene. I was thinking that these paths are all equivalent in the sense that the potential difference along all these paths should be equal to each other. This can give rise to a variational principle. However, probably that is not the case; along each different path, the potential difference is infinite. But we cannot say infinite=infinite. May be a point close enough to the charge will satisfy the condition.

My understanding is you can't even do the comparison because the two end points is undefined. That is the reason people only talk about potential at some distance from the charge. The formula is set up only for points some distance from the charge.

Everyone knows that the potential difference between two points 1 and 2 is given by $$\int_{1}^{2}{\vec{E}}\cdot d\vec{l}$$. My question is that can we calculate the potential difference between two point charges by this formula? Say, one of them is +ve charge and the other is a -ve charge.

Surely the end points give divergence. However, excluding those end points, everything is fine between those two points. Can the potential be defined in some limit?

-Neel

This is why the potential is often defined to be 0 at infinity, so the potential at other
points will be an integral from that point to infinity.

Everyone knows that the potential difference between two points 1 and 2 is given by $$\int_{1}^{2}{\vec{E}}\cdot d\vec{l}$$. My question is that can we calculate the potential difference between two point charges by this formula? Say, one of them is +ve charge and the other is a -ve charge.

Surely the end points give divergence. However, excluding those end points, everything is fine between those two points. Can the potential be defined in some limit?

-Neel

Hi Neel

it is fairly simple to see what happens in the specific case of two generic point charges. Place the reference in the midpoint with both charges on the x-axis (say a and -a are the abscissae of the charges). Consider the path going from one charge to the other along the x-axis. The electric field is conservative in every point but at the extremes so you can calculate your integral on this segment by the variation of the potential at the extremes. Calculate the electric potential in two points (x and -x) symmetric wrt to the origin, calculate the difference and try the limit for x->a . You will see that unless the charges are identical the potential diverges.
I am not sure this limit is independent of the path you choose but this should give you a good starting point...