Question

How do you know that a volt per metre is the same as a newton per coulomb?

Answer 1

Both of these units are used to define the quantity of electric field. Hence, it is important to define the quantity electric field and then derive the similarity between the two units used to define the quantity electric field.

Complete answer:
When a charge is present in space, it exhibits a sphere of influence around it such that force is exerted by it on any other charge within the sphere, varying with distance in inverse-square proportionality (which means that force is inversely proportional to square of distance of separation from the charge).
If any charge $Q$ is present in vicinity of another charge $q$ and the force experience by this charge is F, the electric field by charge $q$ is given by –
$\Rightarrow E = \dfrac{F}{Q}$
Since, the unit of force is newton (N) and that of charge is coulomb (C), we have the unit of electric field as –
$\Rightarrow E = \dfrac{{1N}}{{1C}} = N{C^{ - 1}}$
There is another quantity associated with the electric field known as electric potential.
When a charge $q$ is brought to an electric field of charge $Q$, there is a force of repulsion acting on the charge q. However, if work $W$is done against the electric field to move it through a distance in the field, the work done per unit charge is termed by the quantity electric potential.
Electric potential,
$\Rightarrow V = \dfrac{W}{q}$
Given that, work is the product of force and displacement, $W = F \times d$, we get –
$\Rightarrow V = \dfrac{{F \times d}}{q}$
From the above relation of electric field, we have –
$\Rightarrow V = E \times d$
$\Rightarrow E = \dfrac{V}{d}$
Since, the unit of electric potential is volt (V) and that of displacement is metre (m), we have the unit of electric field as –
$\Rightarrow E = \dfrac{{1V}}{{1m}} = V{m^{ - 1}}$
Therefore, we see that both the units, newton-per-coulomb $N{C^{ - 1}}$ and volt-per-metre $V{m^{ - 1}}$ are the same and both are used interchangeably to define the quantity of electric field.

Note: Even though the two units, newton-per-coulomb $N{C^{ - 1}}$ and volt-per-metre $V{m^{ - 1}}$ are used to define electric field, the former unit, newton-per-coulomb is the fundamental unit and the latter unit, volt-per-metre is the derived unit.