Objectives:

• To understand and appreciate the Coulomb’s Law  $F = \frac{qQ}{4 \pi \epsilon_{0} r^{2}}$  for the force between two point charges
• To be able to apply Coulomb’s Law for the force between two point charges
• To be able to derive and understand electric field strength  $E = \frac{Q}{4 \pi \epsilon_{0} r^{2}}$  for a point charge
• To be able to calculate electric the electric field strength due to both point charges and parallel plates
• To identify similarities and differences between the gravitational field of a point mass and the electric field of a point charge
• To appreciate the concept of electric fields as being one of a number of forms of field giving rise to a force

The electrostatic force

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 (the electrostatic force), and it is expressed as:

$F = \frac{k q_{1} q_{2} }{r^{2}}$

Where;

F = force on each charge (N)
Q1 and Q2 are the interacting charges
r = separation of the charges (m)
The k is a constant of proportionality (like G in Newton’s Law of Universal Gravitation) and is equal to  $k = \frac{1}{4 \pi \epsilon_{0}}$

More traditionally, Coulomb’s law is written as:

$F = \frac{q_{1} q_{2} }{4 \pi \epsilon_{0} r^{2}}$

Where:

$\epsilon_{0}$  is known as the “permittivity of free space”; and is equal to;

$\epsilon_{0} = 8.85 \times 10^{-12} \ C^{2} N^{-1} m^{-2}$

Different materials have different permittivities, and so the permittivity value can be altered in Coulomb’s law for a particular material, but this will be focussed on when studying the capacitors topic.

Important points regarding Coulomb’s law

• The form is exactly the same as Newton’s Law of Universal Gravitation; in particular, it is an inverse-square law;  $F \propto \frac{Q_{1} Q_{2}}{r^{2}}$
• This force can be attractive or repulsive. The magnitude of the force can be calculated by the equation above, 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). Personally, I would suggest that you ignore the signs of the charges in the equation but make sure you draw a diagram showing the electric field, use the diagram to then determine the direction of the force.
• 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 (so a point charge).
• In all realistic cases, the electric force between 2 charged objects absolutely dwarfs the gravitational force between them.

Electric Field strength
The electric field strength is define as the force per unit positive charge;

$E = \frac{F}{Q}$

This equation tells us that the electric field due a to a charge Q will influence a charge q in its field, so the equation can be written as:

$E_{Q} = \frac{F}{Q}$     (1)

Which charge q in the field, the two charged particles will exert an electrostatic force on one another as shown by Coulomb’s law:

$F = \frac{q Q}{4 \pi \epsilon r^{2}}$     (2)

Subbing equation (2) into equation (1) gives:

$E_{Q} = \frac{q Q}{q \times 4 \pi \epsilon r^{2}}$

Cancelling  $q$  gives:

$E_{Q} = \frac{Q}{4 \pi \epsilon r^{2}}$

This is an equation tells us that the electrostatic force varies depending on the distance from the charge itself. This should make sense because in either of the image below;

The closer to the charge, the closer the fields lines are and so the stronger the field. The further away from the charged particle the further apart the fields lines are so the weaker the electric field strength.

Important notes:

• The field strength is a property of the field and not the particular charge that is placed there. For example, at a point where the field strength is 2000 N C-1, a 1 C charge would feel a force of 2000 N whereas a 1 mC charge would feel a force of 2 N; the same field strength, but different forces due to different charges.
• The field strength is a vector quantity. By convention, it points in the direction that a positive charge placed at that point in the field would feel a force.