Question

Derive the relation between electric field and electric potential due to a point charge.

Answer 1

Hint: The electric field due to a point charge at a point is the force acting on a charged body of unit charge placed at that point. The electric potential due to a point charge is the external work done to move a unit charge against the electric field by an infinitesimal distance without any acceleration.

Formula used:
Work done $W$ by a force $F$ on a body is

$W=Fs$

where $s$ is the displacement of the body in the direction of the force.
The electric field $E$ due to a point charge at a point is the force $F$ per unit charge acting on a charged body of charge $q$ placed at that point.

$E=\dfrac{F}{q}$

Complete step by step answer:
To derive the relation, we will consider a positive point charge that produces an electric field $E$ at a point where a positive unit charge is placed.

We will attempt to move the unit charge by an infinitesimal distance $dx$ away from the point charge without any acceleration. Hence, an external force $F$ must be applied that is equal in magnitude and opposite in direction to the force due to the electric field.
Hence, the displacement and the external force are in opposite directions.

Work done $W$ by a force $F$ on a body is

$W=Fs$ --(1)

where $s$ is the displacement of the body in the direction of the force.
Using (1), we get, the work done $dW$ by the external force as

$dW=-Fdx$ --(2)

The negative sign is because the external force and displacement are opposite in direction.
The electric field $E$ due to a point charge at a point is the force $F$ per unit charge acting on a charged body of charge $q$ placed at that point.

$E=\dfrac{F}{q}$ --(3)

Hence, from the explanation above, since the charge at the point is a unit charge, putting this information in equation (3), we will get,

$E=\dfrac{F}{1}=F$ --(4)

Putting (4) in (2), we get,

$dW=-Edx$ --(5)

The electric potential due to a point charge is the external work done to move a unit charge against the electric field by an infinitesimal distance without any acceleration.
Using this definition, we get that the change in electric potential $dV$ due to the point charge will be

$dV=-Edx$ -[Using(5)]
$\therefore E=-\dfrac{dV}{dx}$

Hence, the electric field due to a point charge is the negative of the potential gradient due to the point charge.

Note: Students often get confused and miss out the part in the definition of electric potential that the charge is moved without acceleration. However, it is very important, since this condition makes it valid that the external force can be equated in magnitude with the force exerted by the electric field produced by the point charge. Hence, students must include this condition while writing the definition and deriving the relation and must not forget it.