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 = 90^{o}.

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

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