# Confused about electric potential and electric field

I get easily confused when asked questions about electric field. One question that is bothering me is...

Is it possible to have 0 electric field at a point, but at the same point have electric potential? I've read that the answer is yes, there can be a potential without a electric field, but I can't conceptually understand that. In my mind, if you have a charge sitting between 2 positive charges (0 net electric field), the charge in the middle would have no ability to go anywhere, since there isn't an electric field to move it. Therefore, the potential would be zero. Where is my thinking wrong?

andrewkirk
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Electric potential is a scalar field. Like temperature, it assigns a real number (scalar) to every point in space.

Electric field is a vector field. Like wind, it assigns a vector, which has magnitude and direction, to every point in space.

The absolute level of potential is meaningless. All that has a physical meaning is the difference in potential between two points. So if we have a potential field in which the potential is p at point x, we can make the potential 0 at x by defining a new potential field that gives the potential at any point as the potential under the original potential field, minus p. That new potential field is physically indistinguishable from the original one.

A 2Dimensional analogy that might help you is to think of potential as height above sea level and vector field as slope of the ground. The steeper the ground the stronger the field. If a marble is sitting at the bottom of a round hollow the ground will have no slope so the vector field is zero. The marble won't roll. But it could still be a long way above sea level. Also, it doesn't make any difference to the vector field (slope of the ground) whether we measure potential as height above the low tide sea level or the high tide sea level.

• justin15501 and nasu
Electric potential is a scalar field. Like temperature, it assigns a real number (scalar) to every point in space.

Electric field is a vector field. Like wind, it assigns a vector, which has magnitude and direction, to every point in space.

The absolute level of potential is meaningless. All that has a physical meaning is the difference in potential between two points. So if we have a potential field in which the potential is p at point x, we can make the potential 0 at x by defining a new potential field that gives the potential at any point as the potential under the original potential field, minus p. That new potential field is physically indistinguishable from the original one.

A 2Dimensional analogy that might help you is to think of potential as height above sea level and vector field as slope of the ground. The steeper the ground the stronger the field. If a marble is sitting at the bottom of a round hollow the ground will have no slope so the vector field is zero. The marble won't roll. But it could still be a long way above sea level. Also, it doesn't make any difference to the vector field (slope of the ground) whether we measure potential as height above the low tide sea level or the high tide sea level.
Ok, so the only thing that matters when it comes to electric potential is the difference in potential, meaning I would have to pick another point in space. I think I'm begging to get a grasp. Could you explain the answer of this problem to me?
https://gyazo.com/0e3c73a7d4f6eca57398a2027e32ef5e
If electric potential is a scalar as you mentioned, how do they "cancel out" and make the V=0 depending on the magnitude and the charge?

andrewkirk
Homework Helper
Gold Member
The problem is poorly written. We can infer that the two charges are of equal magnitude by the symmetry of the diagram (assuming that the levels of contours are equidistant), but we can infer nothing about the signs. Consider if they were contour lines of altitude rather than of charge. The diagram would look the same regardless of whether the two points were both the tops of hills or the bottoms of pits, or one of each. We can't tell which without the contour lines being labelled with level values.

I am guessing that when they write V=0 they actually meant to write that the electric field is zero. If that's what they meant then the answer is that the charges have the same sign. But in that case the question is wrongly written.

Alternatively, they might have meant V=0 is the lowest potential on the diagram, ie that every other point on the diagram has a higher potential. That would also lead to a conclusion that the signs are the same. But I am not aware of any convention that would enable them to expect students to make that interpretation of the question.

If electric potential is a scalar as you mentioned, how do they "cancel out" and make the V=0 depending on the magnitude and the charge?

They use infinity as the point with zero potential. If you integrate the force along a path to infinity you get the potential. If you do this, it's possible to calculate the potential due to 2 or more point charges as the sum of the potentials, because the integral of the sum of the forces is the same as the sum of the integrals.

They use infinity as the point with zero potential. If you integrate the force along a path to infinity you get the potential. If you do this, it's possible to calculate the potential due to 2 or more point charges as the sum of the potentials, because the integral of the sum of the forces is the same as the sum of the integrals.
So if you have two point charges that each have there own potential, like Qa and Qb in the example, I still don't get how they can be zero. You the two potentials together as you said. The only way i could see them canceling out is if one potential is a negative and one is a positive, then it would be V=0. But since potential is a scalar and not a vector, you can't have negative potential because the negative would be the direction? Thanks for the help so far!

So if you have two point charges that each have there own potential, like Qa and Qb in the example, I still don't get how they can be zero. You the two potentials together as you said. The only way i could see them canceling out is if one potential is a negative and one is a positive, then it would be V=0. But since potential is a scalar and not a vector, you can't have negative potential because the negative would be the direction? Thanks for the help so far!

Potential has a sign, but not a direction. A positive potential means that a positive charge has more potential energy, then if it was at infinity, and a negative sign means it has less potential energy. If you have a charge that would have a potential energy E1 at a distance d1 from a point charge Q1, and an energy E2 at a distance d2 from a point charge Q2, it will have an energy E1+E2, if both charges are present.
If Q1 and Q2 have opposite signs and d1 and d2 are equal, the potential energy is 0.

The problem is poorly written. We can infer that the two charges are of equal magnitude by the symmetry of the diagram (assuming that the levels of contours are equidistant), but we can infer nothing about the signs. Consider if they were contour lines of altitude rather than of charge. The diagram would look the same regardless of whether the two points were both the tops of hills or the bottoms of pits, or one of each. We can't tell which without the contour lines being labelled with level values.

I am guessing that when they write V=0 they actually meant to write that the electric field is zero. If that's what they meant then the answer is that the charges have the same sign. But in that case the question is wrongly written.

Alternatively, they might have meant V=0 is the lowest potential on the diagram, ie that every other point on the diagram has a higher potential. That would also lead to a conclusion that the signs are the same. But I am not aware of any convention that would enable them to expect students to make that interpretation of the question.
I'm confused myself about your first reply. I still struggle to understand why you can have an electric field of zero, but still have a potential. This is the statement you offered:

"The absolute level of potential is meaningless. All that has a physical meaning is the difference in potential between two points. So if we have a potential field in which the potential is p at point x, we can make the potential 0 at x by defining a new potential field that gives the potential at any point as the potential under the original potential field, minus p. That new potential field is physically indistinguishable from the original one."

Correct my thinking if it is wrong about an area where there is zero electric field:
Is there a potential? Yes, because potential is relative and depending on the other point you pick, it could have a value.
Is there potential energy? No, because in the middle of an electric field there is no field to move it.

Also, is potential energy a vector or a scalar?

Also why is the equation for potential energy = kq1q2/r^2 the same thing as the coulombs law equation?

andrewkirk
Homework Helper
Gold Member
Also, is potential energy a vector or a scalar?
It's a scalar. It's equal to the potential of that point in space multiplied by the charge of the particle at that point.

Also why is the equation for potential energy = kq1q2/r^2 the same thing as the coulombs law equation?
It's not. As I recall, the PE is ##\frac{kq_1q_2}{r}## whereas the Coulomb force magnitude (because the force is a vector) is ##\frac{kq_1q_2}{r^2}##. The reason for the similarity is that the electric field is the vector gradient of the scalar potential field. Multiply both by the charge of the particle and you get the Coulomb force and the PE respectively.

It's a scalar. It's equal to the potential of that point in space multiplied by the charge of the particle at that point.

It's not. As I recall, the PE is ##\frac{kq_1q_2}{r}## whereas the Coulomb force magnitude (because the force is a vector) is ##\frac{kq_1q_2}{r^2}##. The reason for the similarity is that the electric field is the vector gradient of the scalar potential field. Multiply both by the charge of the particle and you get the Coulomb force and the PE respectively.
Well that was a silly mistake on my part. Thanks for the clarification. Could you explain this vector gradient thing a little bit more? Our professor completely brushed over it because most the class has only taken Calc 2 and doesn't know about partials yet, which is the boat I am in. I think the reason I'm so confused about the relationship between electric field and electric potential is because I don't understand gradient.