Specified a set of equipotential surfaces how does one determine the corresponding set of electric field lines?
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An equipotential surface signifies a set of points all having the same value of electric potential that is the same value of electric potential energy per charge. That denotes that if you were to move a test charge from one point on an equipotential surface to another it would experience no change in electric potential energy. This denotes that the electric field the test charge is in would do no work on the test charge since work = the negative of the change in potential energy. Currently work is forcealong the path times the length of the path and the electric field is force per charge. The only way you are able to move a charged particle in an electric field from one point to another with no work being done on the particle by the electric field along any part of the path is for the force-along-the-path to be zero. In this case this signifies that the component of the electric field along the path must be zero. This denotes that the electric field must be perpendicular to the path. Since the path be able to be any path in the equipotential surface the electric field must everywhere be perpendicular to the equipotential surface. This argument holds factual for any equipotential surface so the electric field is perpendicular to all equipotential surfaces.