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Question
Explain why, for a charge configuration, the equipotential surface through a point is normal to the electric field at that point
Solution
For any charge configuration, the equipotential surface through a point is normal to the electric field at that point. If the field was not normal to the equipotential surface, it would have a non-zero component along the surface. To move a unit test charge against the direction of the component of the field, work would have to be done. However, this is in contradiction to the definition of an equipotential surface: there is no potential difference between any two points on the surface and no work is required to move a test charge on the surface. Therefore, the electric field must be normal to the equipotential surface at every point.
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