Question

Mark the correct statement:
A. If $E$ is zero at a certain point then $V$should be zero at that point.
B. If $E$ is not zero at a certain point then $V$ should not be zero at that point.
C. If $E$ is not zero at a certain point then $V$ should be zero at that point.
D. If $V$ is zero at a certain point then $E$ may or may not be zero at that point.

Answer 1

The relation between the electric field and the potential is established and hence the equation relating them would specify whether potential depends on the electric field. The consideration of the conditions wherein each of given cases is valid, must be tested in order to select the correct statement.

Complete step by step answer:
The above problem revolves around the concept of electric field and potential and relationship between them. Let us first understand their concepts. Electric field is defined as a field that exists at a point or a region if a force is exerted on a stationary charge particle placed at that point. It is the field that is set up due to the electrostatic force between two charged bodies separated by a distance, that is, they would not have to be in contact.

The electrostatic potential, however, is the amount of work done in moving a unit positive charge from infinity, that is, from a far-away distance to a point against the electrostatic force.
The relation between potential and the electric field is given below:
$V = Er$
Where, $V$ is the potential, $E$ is the electric field and $r$ is the distance of separation.
By rearranging the terms to make electric field as the subject we get:
$E = \dfrac{V}{r}$
When we rewrite the same equation in terms of differentiation wherein a small amount of potential is differentiated with respect to the distance $d$ then we have:
$E = - \dfrac{{dV}}{{dr}}$ ------($1$)
Hence, the electric field is equivalent to the rate of change of potential with distance and hence it is known as the potential gradient. The negative sign represents the direction of the electric field which is in the direction of decreasing potential.

Thus, it can be seen from the relation from equation ($1$) that since electric field is just the differentiation of potential and hence if electric field is zero at some point then this means that the potential value must be either equivalent to zero or a constant value accordingly. Hence, it can be concluded that the first option A) would be incorrect because potential need not be only zero necessarily as it can take the value of a constant as well. Option B) is also incorrect because when electric field is not zero then potential might be zero as in the case of equipotential surfaces where a component of electric field is present and hence it is not zero.Similarly when potential is zero then the electric field may more may not be zero and hence option B is also incorrect.

Also, another important point is that the electric field is a vector quantity and potential is a scalar quantity and hence electric field not only depends on its magnitude but also depends on the direction of the field. Hence, even when potential is zero the electric field vector might have some component of it in the horizontal or vertical direction and thus it is not necessary that electric field must be zero when the potential is zero. Hence, option D) would be the correct option. Therefore, the potential , $V$, is zero at a certain point but the electric field, $E$, may or may not be zero at that point.

Thus, the correct option is option D.

Note: The electric field is in the direction in which the potential decrease is steepest. This is one of the properties of the relation between electric field and potential. There may be a misconception that when potential is zero then in accordance with the formula relating $E$ and $V$, the electric field is also assumed to be zero which is wrong.