Coulomb's Law

Between 1785 and 1787, Charles Augustine de Coulomb performed a series of experiments involving electric charges, and eventually established what is nowadays known as Coulomb's law. According to this law, any two point electric charges (i.e., electrically charged objects of negligible spatial extents) exert a force on one another. This force is directed along the line of centers joining the two charges, is repulsive for two like charges and attractive for opposite charges, is directly proportional to the product of the charges, and is inversely proportional to the square of the distance between the charges.

Consider a system consisting of two point electric charges. Let charge 1 have electric charge $q_1$ and displacement ${\bf r}_1$. Let charge 2 have electric charge $q_2$ and displacement ${\bf r}_2$. Coulomb's law states that the electrical force exerted on charge $2$ by charge $1$ is

$${\bf f}_{21} = \frac{q_1\, q_2}{4\pi\,\epsilon_0} \frac{{\bf r}_2 - {\bf r}_1} {\vert{\bf r}_2-{\bf r}_1\vert^{3}}.$$ (2.2)

An equal and opposite force acts on the first charge, in accordance with Newton's third law of motion. (See Section 1.2.4.) The universal constant $\epsilon_0$ is called the electrical permittivity of free space, and takes the value

$$\epsilon_0 = 8.854\times 10^{-12} \,\,{\rm C^{\,2} \,N^{-1}m^{-2}}.$$ (2.3)

As we saw in Section 1.8.1, according to Newtonian gravity, if two point mass objects of masses $m_1$ and $m_2$ are located at displacements ${\bf r}_1$ and ${\bf r}_2$, respectively, then the gravitational force acting on the second object is

$${\bf f}_{21} = - G \,m_1 \,m_2\, \frac{{\bf r}_2 - {\bf r}_1} {\vert{\bf r}_2 - {\bf r}_1\vert^{3}}.$$ (2.4)

The universal gravitational constant $G$ takes the value

$$G = 6.674\times 10^{-11}\,\,{\rm N \,m^2 \,kg^{-2}}.$$ (2.5)

[See Equation (1.239).] Note that Coulomb's law has the same mathematical form as Newton's law of gravity. In particular, they are both inverse-square force laws; that is,

$$\vert{\bf f}_{21}\vert \propto \frac{1}{\vert{\bf r}_2- {\bf r}_1\vert^2}.$$ (2.6)

However, Coulomb's and Newton's laws differ in two crucial respects. First, the force due to gravity is always attractive (because there is no such thing as a negative mass). Second, the magnitudes of the forces predicted by the two laws are vastly different. Consider the ratio of the electrical and gravitational forces acting on two particles. This ratio is a constant, independent of the relative positions of the particles, and is given by

$$\frac{\vert{\bf f}_{\rm electrical}\vert}{\vert{\bf f}_{\rm gravi... ...{\vert q_1\vert}{m_1}\frac{\vert q_2\vert}{m_2} \frac{1}{4\pi\,\epsilon_0 \,G}.$$ (2.7)

For electrons, the charge to mass ratio is $\vert q\vert/m = 1.759\times 10^{11}~{\rm C\,kg^{-1}}$, so

$$\frac{\vert{\bf f}_{\rm electrical}\vert}{\vert{\bf f}_{\rm gravitational}\vert} = 4.17\times 10^{42},$$ (2.8)

which is a truly colossal number. Suppose we were studying a physics problem involving the motion of particles under the action of two forces with the same spatial range, but differing in magnitude by a factor $10^{42}$. It would seem a plausible approximation (to say the least) to start the investigation by neglecting the weaker force altogether. Applying this reasoning to the motion of particles in the universe, we would expect the universe to be governed entirely by electrical forces. However, this is not the case. The force that holds us to the surface of the Earth, and prevents us from floating off into space, is gravity. The force that causes the Earth to orbit the Sun is also gravity. In fact, on astronomical lengthscales, gravity is the dominant force, and electrical forces are largely irrelevant. The key to understanding this paradox is that there are both positive and negative electric charges, whereas there are only positive gravitational “charges.” This implies that gravitational forces are always cumulative, whereas electrical forces can cancel one another out. Suppose, for the sake of argument, that the universe starts out with randomly distributed electric charges. Initially, we expect electrical forces to completely dominate gravitational forces. These forces act to cause every positive electric charge to get as far away as possible from the other positive charges in the universe, and as close as possible to the other negative charges. After a while, we would expect the positive and negative electric charges to form close pairs. Just how close is determined by quantum mechanics, but, in general, it is fairly close; that is, about $10^{-10}$ m. The electrical forces due to the charges in each pair effectively cancel one another out on lengthscales much larger than the mutual spacing of the pair. However, it is only possible for gravity to be the dominant long-range force in the universe if the number of positive electric charges is almost equal to the number of negative charges. In this situation, every positive charge can find a negative charge to team up with, and there are virtually no charges left over. In order for the cancellation of long-range electrical forces to be effective, the relative difference in the number of positive and negative electric charges in the universe must be incredibly small. In fact, positive and negative charges have to cancel one another to such accuracy that most physicists believe that the net electric charge of the universe is exactly zero. But, it is not sufficient for the universe to start out with zero net charge. Suppose there were some elementary particle process that did not conserve electric charge. Even if this were to go on at a very low rate, it would not take long before the fine balance between positive and negative charges in the universe was wrecked. Thus, it is important that electric charge is a conserved quantity (i.e., the net charge of the universe can neither increase or decrease). As far as we know, this is the case. To date, no elementary particle reaction has been discovered that can create or destroy net electric charge.

In summary, there are two long-range forces in the universe, electricity and gravity. The former is enormously stronger than the latter, but is usually hidden away inside neutral atoms. The fine balance of forces due to negative and positive electric charges starts to break down on atomic scales. In fact, interatomic and intermolecular forces are all electrical in nature. So, electrical forces are basically what prevent us from falling though the floor. But, this is electromagnetism on the microscopic, or atomic, scale. Classical electromagnetism generally describes phenomena in which some sort of violence is done to matter, so that the close pairing of negative and positive electric charges is disrupted, allowing electrical forces to manifest themselves on macroscopic lengthscales. Of course, very little disruption is necessary before gigantic forces are generated. Hence, it is no coincidence that the vast majority of useful machines that humankind has devised during the last century or so are electrical in nature.