Question

For any charge configuration, the equipotential surface through a point is _____ to the electric field at a point.

Answer 1

The relation between equipotential surface and the electric field at a point can be easily understood by using the Work done formula for a moving test charge and the angle between them.

Complete step by step answer:
As the name suggests, an equipotential surface is a surface in space where the potential in every point of that plane is the same. Apart from being a scalar potential, equipotential concepts can also be applied in vector concepts. Electric field on the other hand is described as the physical field or region around a particle which is charged and exerts a force on other neighbouring charges on the field or region. The force can be attractive or repulsive.

We know that the formula for work done by a moving test charge is $W=Fs\cos \theta$ , where $s$ is the magnitude of displacement and $F$ is the electric force. $\theta$ is the relation we are looking for. We know that in an equipotential surface or plane, the work done is zero in moving a test charge. So,
$W=Fs\cos \theta =0 \\\ \Rightarrow \cos \theta =0 \\\ \therefore \theta ={{90}^{\circ }}$ , as $F$ and $s$ cannot be zero.
By using the formula of work done we can easily see that the angle between the equipotential surface and the electric field at a point is perpendicular.

Therefore, For any charge configuration, the equipotential surface through a point is ” perpendicular” to the electric field at a point.

Note: If you want to find the direction of the electric force, it must be known that the direction of the electric field lines shows the direction of the electric force on that charge.If the points in an electric field are all at the same electric potential, then they are known as the equipotential points.