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Coulomb's Law of force between charges

The force between two charges Q1 and Q2 separated by a distance d is expressed by Coulombs law, proposed in 1785:

Coulomb's law for the force (F) between two charges: F = (1/4pe)Q1Q2/d2

where e is a constant for a given material known as the permittivity of that material. Basically changing e changes the force between the two charges, for example if e is large the force is small.

In a vacuum this equation becomes

Coulomb's law for the force (F) between two charges in a vacuum: F = (1/4peo)Q1Q2/d2

where eo is the permittivity of free space (8.84x10-12 Fm-1)

A useful quantity in calculations is 1/4peo - this is equal to 9x109 F m-1

Notice that this force is:
(a) proportional to 1/d2
(b) dependent on the electrical properties of the material between the charges (e)
(c) it may be attractive or repulsive
It is often useful to compare the electrostatic properties of material by using a quantity known as the relative permittivity (er). This is defined using the relation:

e = eoer

Some examples of relative permittivities (at 293 K) are given in the following table:

Solid     Liquid     Gas  
Amber 2.8   Castor oil 4.5   Air 1.000 0536
Ebonite 2.8   Glycerol 43   Carbon dioxide 1.000 986
Glass 5-10   Ethoxyethane 4.3   Helium 1.000 07
Mica 5.7-6.7   Nitrobenzene 35.7   Hydrogen 1.000 27
PVC 4.5   Paraffin 2.2   Nitrogen 1.000 580
Polystyrene 2.55   Propanone 21.3   Oxygen 1.000 53
Teflon 2.1   Turpentine 2.23   Water vapour (393 K) 1.000 60
Wax 5-10   Water 80.4      

The force between two point charges and between two charged plates

(a) two point charges
this can be done with two light balls suspended on threads. They are given charges of equal sign and repel each other. By measuring the angle between the threads and the masses of the balls the force between them can be found and the variation of force with charge can be investigated.
It is interesting to discuss what would happen if the masses of the two balls were the same but the charges on them were different. Would the threads make the same angle with the vertical?

(b) two parallel plates
The formula for the force between two plates is F = [QE] = [QV/t] and using the relation for the capacitance of a parallel plate capacitor of area A and separation t (C = Q/V = eoA/t) the force between two charged plates is:

F = [eoAV2/t2]

It is very simple to test this using a top pan balance and two aluminium plates.

Example problems
1. Calculate the force between two charges of 10-5 C separated by a distance of 0.2 m in a vacuum. Force (F) = [1/4peo][Q2/d2] = [9x109x10-10]/0.22 = 2.5 N

2. Calculate the value of two similar charges separated by 0.5 m in a vacuum if the force between them is 1.5 N.

Q2 = [4peo][Fd2] = [1/9x109]x[1.5x0.25] = 4.17x10-11
Q = 6.45x10-6 C
© Keith Gibbs 2011