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

A volt per meter (V/m) and a newton per coulomb (N/C) are equivalent units of measurement and represent the same physical quantity. This equivalence arises from the fundamental relationships between electric field (measured in volts per meter) and electric force (measured in newtons per coulomb) in the context of electrostatics.

Here's why they are equivalent:

Electric Field (E) in Volts per Meter (V/m): Electric field is a measure of the force per unit charge experienced by a charged particle in an electric field. It's defined as the electric force (F) acting on a charged object divided by the magnitude of the charge (q) of that object. Mathematically, it can be expressed as:

E = F / q

The SI unit of electric field is the volt per meter (V/m). In other words, if you apply a voltage of 1 volt across a distance of 1 meter, you create an electric field of 1 V/m.

Electric Force (F) in Newtons (N): The electric force between two charged objects is governed by Coulomb's law, which states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

F = k * (q1 * q2) / r^2

Where:

F is the electric force in newtons (N).
q1 and q2 are the magnitudes of the charges in coulombs (C).
r is the distance between the charges in meters (m).
k is Coulomb's constant, a proportionality constant.
Now, to show that V/m and N/C are equivalent:

Using the definition of electric field (E = F / q), you can rearrange it as follows:

F = E * q

Substitute this into Coulomb's law:

E * q = k * (q1 * q2) / r^2

Cancel out the charge 'q' on both sides:

E = k * (q1 * q2) / (r^2 * q)

Now, we have the electric field (E) expressed in terms of charges (q1 and q2) and distance (r), and we know that electric field is measured in V/m.

From the above equation, it's clear that E is in V/m, but it's also equal to k * (q1 * q2) / (r^2 * q), which has the units N/C.

So, a volt per meter (V/m) is indeed equivalent to a newton per coulomb (N/C) because they both represent the same physical quantity: the electric field strength. This equivalence is fundamental to the way we describe and understand electric fields and electric forces in the context of electrostatics.