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## Main Question or Discussion Point

Given n (finite) point charges in the xy-plane, is it possible to have a curve (in the plane) along which the electrostatic force vanishes (F=0)?

I know that it's possible to have a curve through space along which the force vanishes when all of the charges are in the plane. For instance, place an even number of point charges of equal magnitude but alternating sign equidistantly on a circle. Then the line orthogonal to the plane of the charges passing through the center of the circle will have net force of 0 at every point.

Is it possible to generate such a curve in the plane, when all of the points are located in the plane?

I have already proven that if such a curve were to exist, the potential along this curve must also be zero (I know this result is true if all of the charges have the same sign...I think it's true in every case). Also, I know that such a curve cannot exist if all of the charges have the same sign. I just don't know if this result extends to any configuration of positive and negative charges.

I know that it's possible to have a curve through space along which the force vanishes when all of the charges are in the plane. For instance, place an even number of point charges of equal magnitude but alternating sign equidistantly on a circle. Then the line orthogonal to the plane of the charges passing through the center of the circle will have net force of 0 at every point.

Is it possible to generate such a curve in the plane, when all of the points are located in the plane?

I have already proven that if such a curve were to exist, the potential along this curve must also be zero (I know this result is true if all of the charges have the same sign...I think it's true in every case). Also, I know that such a curve cannot exist if all of the charges have the same sign. I just don't know if this result extends to any configuration of positive and negative charges.

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