Question

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

Answer 1

Consider two points $A$ and $B$ separated by a small distance $dx$ in an electric field.

Since $dx$ is small, the electric field $E$ is assumed to be uniform along $AB$. The force acting on a unit positive charge at A is equal to $E$.

Now, the work done in moving a unit positive charge from $A$ to $B$ against the electric field is $dW=-E\times dx$.

The negative sign shows that the work is done against the direction of the field. Since the work done is equal to potential difference $dV$ between $A$ and $B$ then $dV=-E\times dx$ or $E=-\frac{dV}{dx}$

Thus, the electric field at a point is the negative of the potential gradient at that point.