Question

for any charge configuration equipotential surface through a point and normal to the electric field justify

Answer 1

We know that the work done(W) in moving a test charge along an equipotential surface is zero and this is because an equipotential surface is a surface with a constant value of potential at all points on the surface.

where, F = Electric force
            s = Magnitude of displacement of the charge
For non-zero displacement this is possible only when cos thita =0 , where thita =90°
Thus, the force acting on the point charge is perpendicular to the equipotential surface.
We also know that the lines of force or the electric field lines indicate the direction of electric force on a charge, therefore, for any charge configuration, euipotential surface through a point is normal to the electric field.

Answer 2

Hey !!

If the electric field were not normal to the surface, then it would have a component along the surface which would cause work to be done in moving a charge on an equipotential surface.

DETAILED EXPLANATION

Equipotential lines are curved lines with similar altitude that pertains to electric potential or voltage. These lines are always perpendicular to electric field which creates equipotential surfaces in three dimensions. The movement along equipotential surface requires no work as movement is perpendicular to electric field.

As work done in moving test charge along equipotential surface is zero, there appear fixed potential values across every point on equipotential surface so,

           W = Fcosθ = 0

where,

                     F = Electric force

s = Magnitude of displacement of charge

In case, of non-zero displacement, it exists only when

             cos θ = 0

                    θ = 90°

Hence, force acting on point charge is perpendicular to equipotential surface.

Good luck !!