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1.5: Coulomb's Law

If you are interested in the history of physics, it is well worth reading about the important experiments of Charles Coulomb in 1785. In these experiments he had a small fixed metal sphere which he could charge with electricity, and a second metal sphere attached to a vane suspended from a fine torsion thread. The two spheres were charged and, because of the repulsive force between them, the vane twisted round at the end of the torsion thread. By this means he was able to measure precisely the small forces between the charges, and to determine how the force varied with the amount of charge and the distance between them.

From these experiments resulted what is now known as Coulomb’s Law.  Two electric charges of like sign repel each other with a force that is proportional to the product of their charges and inversely proportional to the square of the distance between them:

\[F \propto \frac{Q_1Q_2}{r^2}\label{1.5.1}\]

Here \(Q_1\) and \(Q_2\) are the two charges and \(r\) is the distance between them.

We could in principle use any symbol we like for the constant of proportionality, but in standard SI (Système International) practice, the constant of proportionality is written as \(\frac{1}{4\pi \epsilon}\) so that Coulomb’s Law takes the form

\[F=\frac{1}{4\pi\epsilon}\frac{Q_1Q_2}{r^2}\label{1.5.2}\]

Here \(\epsilon\) is called the permittivity of the medium in which the charges are situated, and it varies from medium to medium. The permittivity of a vacuum (or of “free space”) is given the symbol \(\epsilon_0\). Media other than a vacuum have permittivities a little greater than \(\epsilon_0\). The permittivity of air is very little different from that of free space, and, unless specified otherwise, I shall assume that all experiments described in this chapter are done either in free space or in air, so that I shall write Coulomb’s Law as

\[F=\frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{r^2}\label{1.5.3}\]

You may wonder – why the factor 4\(\pi\)? In fact it is very convenient to define the permittivity in this manner, with 4\(\pi\) in the denominator, because, as we shall see, it will ensure that all formulas that describe situations of spherical symmetry will include a 4\(\pi\), formulas that describe situations of cylindrical symmetry will include 2\(\pi\), and no \(\pi\) will appear in formulas involving uniform fields. Some writers (particularly those who favour cgs units) prefer to incorporate the 4\(\pi\) into the definition of the permittivity, so that Coulomb’s law appears in the form \(F=Q_1Q_2/(\epsilon_0r^2)\), though it is standard SI practice to define the permittivity as in equation \ref{1.5.3}. The permittivity defined by equation \ref{1.5.3} is known as the “rationalized” definition of the permittivity, and it results in much simpler formulas throughout electromagnetic theory than the “unrationalized” definition.

The SI unit of charge is the coulomb, C. Unfortunately at this stage I cannot give you an exact definition of the coulomb, although, if a current of 1 amp flows for a second, the amount of electric charge that has flowed is 1 coulomb. This may at first seem to be very clear, until you reflect that we have not yet defined what is meant by an amp, and that, I’m afraid, will have to come in a much later chapter.

Until then, I can give you some small indications. For example, the charge on an electron is about -1.6022 X 10-19 C, and the charge on a proton is about +1.6022 X 10-19 C. That is to say, a collection of 6.24 X 1018 protons, if you could somehow bundle them all together and stop them from flying apart, amounts to a charge of 1 C. A mole of protons (i.e. 6.022 X 1023 protons) which would have a mass of about one gram, would have a charge of 9.65 X 104 C, which is also called a faraday (which is not at all the same thing as a farad).

[The current definition of the coulomb and the amp, which will be given in Chapter 6, requires some knowledge of electromagnetism. However, it is likely that, in 2018, the coulomb will be redefined in such a manner that the magnitude of the charge on a single electron is exactly 1.60217 X 10-19 C.]

The charges involved in our experiments with pith balls, glass rods and gold-leaf electroscopes are very small in terms of coulombs, and are typically of the order of nanocoulombs.

The permittivity of free space has the approximate value

\[\epsilon_0 = 8.8542\times 10^{-12}\text{ C}^2\, \text{N}^{-1}\, \text{m}^{-2}. \nonumber\]

Later on, when we know what is meant by a “farad”, we shall use the units F m-1 to describe permittivity – but that will have to wait until section 5.2.

You may well ask how the permittivity of free space is measured. A brief answer might be “by carrying out experiments similar to those of Coulomb”. However – and this is rather a long story, which I shall not describe here – it turns out that since we today define the metre by defining the speed of light, c, to be exactly 2.997 925 58 108 m s-1, the permittivity of free space has a defined value, given, in SI units, by

\[4\pi\epsilon_0 = \frac{10^7}{c^2}\nonumber\]

It is therefore not necessary to measure \(\epsilon_0\) any more than it is necessary to measure c. But that, as I say, is a long story.

New Definition

[But if, as is likely, the new definition of the coulomb, referred to on the previous page, becomes official in 2018, \(\epsilon_0\) will no longer have an exact defined value, but its measured value will be approximately 8.8542 X 10-12 C2 N-1 m-2. Many teaching laboratories run an undergraduate experiment in which students measure the charge on a capacitor of known physical dimensions and a measured potential difference between the plates, and this enables the measured value of \(\epsilon_0\) to be calculated.]

From the point of view of dimensional analysis, electric charge cannot be expressed in terms of M, L and T, but it has a dimension, Q, of its own. (This assertion is challenged by some, but this is not the place to discuss the reasons. I may add a chapter, eventually, discussing this point much later on.) We say that the dimensions of electric charge are Q.

Exercise \(\PageIndex{1}\)

Show that the dimensions of permittivity are

\[[\epsilon_0]  =\text{M}^{-1}\, \text{L}^{-3}\, \text{T}^2 \,\text{Q}^2\nonumber\]

I shall strongly advise the reader to work out and make a note of the dimensions of every new electric or magnetic quantity as it is introduced.

Exercise \(\PageIndex{2}\)

Calculate the magnitude of the force between two point charges of 1 C each (that’s an enormous charge!) 1 m apart in vacuo.

Solution

The answer, of course, is 1/(4\(\pi \epsilon_0\)), and that, as we have just seen, is c2/107 = 9 X 109 N, which is equal to the weight of a mass of 9.2 X 105 tonnes or nearly a million tonnes.

Exercise \(\PageIndex{3}\)

Calculate the ratio of the electrostatic to the gravitational force between two electrons. The numbers you will need are: \(Q\) = 1.60 X 10-19 C, m = 9.11 X 10-31 kg,  \(\epsilon_0\) = 8.85 X 10-12 N m2 C-2 , G = 6.67 X 10-11 N m2 kg-2 .

Solution

The answer, which is independent of their distance apart, since both forces fall off inversely as the square of the distance, is \(Q^2/(4\pi\epsilon_0Gm^2)\) (and you should verify that this is dimensionless), and this comes to 4.2 X 1042. This is the basis of the oft-heard statement that electrical forces are 1042 times as strong as gravitational forces – but such a statement out of context is rather meaningless. For example, the gravitational force between Earth and Moon is much more than the electrostatic force (if any) between them, and cosmologists could make a good case for saying that the strongest forces in the Universe are gravitational.

The ratio of the permittivity of an insulating substance to the permittivity of free space is its relative permittivity, also called its dielectric constant. The dielectric constants of many commonly-encountered insulating substances are of order “a few”. That is, somewhere between 2 and 10. Pure water has a dielectric constant of about 80, which is quite high (but bear in mind that most water is far from pure and is not an insulator.) Some special substances, known as ferroelectric substances, such as strontium titanate SrTiO3, have dielectric constants of a few hundred.

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