## Episode 407: Coulomb's law

Lesson for 16-19

- Activity time 40 minutes
- Level Advanced

This episode introduces Coulomb’s law, which gives the force between two charges, in exactly the same way that Newton’s Law of Universal Gravitation gives the force between two masses. In fact, we will see that the two laws are identical in structure.

Lesson Summary

- Discussion: Coulomb’s law (15 minutes)
- Worked examples: Calculations involving Coulomb’s law (25 minutes)

#### Discussion: Coulomb’s law

We know that a field exists around a charge that exerts force on other charges placed there, but how can we calculate the force? The force will be dependent upon the sizes of the charges, and their separation. In fact the force follows an inverse square law, and is very similar in form to Newton’s Law of Universal Gravitation. It is known as Coulomb’s law, and it is expressed as:

*F* = *k* *Q*_{1} *Q*_{2}*r*^{ 2}

where *F*is the force on each charge (N)

*Q*_{1} and *Q*_{2} are the interacting charges (C)

*r* is the separation of the charges (m)

The *k* is a constant of proportionality (like *G* in Newton’s Law of Universal Gravitation). In a vacuum, and to all intents and purposes, in air, we have

*k* = 9.0 × 10^{9} N m^{2} C^{-2} (units obtained by rearranging the original equation)

More traditionally, Coulomb’s law is written:

*F* = *Q*_{1} *Q*_{2}4 π ε_{0} *r*^{ 2}

where ε_{0} is known as the permittivity of free space

; ε_{0} = 8.85 × 10^{-12} F m^{-1}
(farads per metre). Permittivity is a property of a material that is indicative of how well it supports an electric field, but is beyond the scope of these notes. Thus, we have
*k* = 14 π ε_{0}. Different materials have different permittivities, and so the value of k in Coulomb’s law also changes for different materials.

Points to bring out about Coulomb’s law:

The form is exactly the same as Newton’s law of universal gravitation; in particular, it is an inverse-square law.

This force can be attractive or repulsive. The magnitude of the force can be calculated by this equation, and the direction should be obvious from the signs of the interacting charges. (Actually, if you include the signs of the charges in the equation, then whenever you get a negative answer for the force, there is an attraction, whereas a positive answer indicates repulsion).

Although the law is formulated for point charges, it works equally well for spherically symmetric charge distributions. In the case of a sphere of charge, calculations are done assuming all the charge is at the centre of the sphere.

In all realistic cases, the electric force between 2 charges objects absolutely dwarfs the gravitational force between them, as the first of the worked examples will show.

#### Worked examples: Calculations involving Coulomb’s law

Episode 407-1: Worked examples – Coulomb’s law (Word, 27 KB)