If electric potential is constant throughout in a region of space what will be the electric field in this region?

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The Electric field is defined as the gradient of the electric potential. The electric potential is a scalar field, that is it has no direction, only measured values at each point.

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

If the electric potential is constant, say 1 V, then there is no direction of greatest increase. Hence the gradient of the electric potential is zero, and the electric field is zero everywhere inside the region of constant electric potential.

Electrical potential is constant, inside an insulated conductor, located in the outside electric field .

So if there is an isolated conductor in an electrostatic field, on its external surface are induced electric charges of different signs and electrostatic equilibrium is achieved in the following conditions:

1. Intensity of the electric field, inside conductor is zero;

2. External surface of the conductor has equivalent potential, which means that the intensity of the electric field, outside the conductor, is perpendicular to it's outer surface.

If an isolated conductor is charged with the electric charge Q, this is distributed on the outer surface of the conductor until the moment when conditions 1 and 2 are satisfied.

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