show that the electric field is always directed perpendicular to an equi-potential surface.

For an equipotential Surface, Potential Difference of any 2 points on it = 0

Hence, Delta V = 0

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We know that, Delta V = E.dl

hence, 0 = E.dl

0 = E x dl x Costheta

here, E cant be 0, and dl also cant be 0.

Therefore, Costheta = 0

Theta = 90o.

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Hence the electric field is always directed perpendicular to an equipotential surface.

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if it is not perpendicular ,the electric field can be resolve into two components ,i.e., one parallel and one perpendicular to the surface. so when a charge q is moved through the surface ,some work is to be done against the parallel electric field component which contradicts the property of equipotential surface and hence electric field should be perpendicular to the equipotential surface.

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