Coulomb's Law

The first precise measurement of the force between two electric charges was performed by the French scientist Charles-Augustin de Coulomb in 1788. Coulomb concluded that:

The electrical force between two charges at rest is directly proportional to the product of the charges, and inversely proportional to the square of the distance between the charges

This law of force is nowadays known as Coulomb's law. Incidentally, an electrical force exerted between two stationary charges is known as an electrostatic force. In algebraic form, Coulomb's law is written

$$f = \frac{q\,q'}{4\pi\epsilon_0\,r^2},$$ (54)

where $f$ is the magnitude of the force, $q$ and $q'$ are the magnitudes of the two charges (with the appropriate signs), and $r$ is the distance between the two charges. The force is repulsive if $f>0$, and attractive if $f<0$. The universal constant

$$\epsilon_0 = 8.854\times 10^{-12}\,\,{\rm N^{-1}\, m^{-2}\,C^2}$$ (55)

is called the permittivity of free space or the permittivity of the vacuum. We can also write Coulomb's law in the form

$$f = k_e\, \frac{q \,q'}{r^2},$$ (56)

where the constant of proportionality $k_e=1/4\pi\epsilon_0$ takes the value

$$k_e = 8.988\times 10^9\,\,{\rm N\,m^2\,C^{-2}}.$$ (57)

Coulomb's law has an analogous form to Newton's law of gravitation,

$$f = -G\,\frac{m\,m'}{r^2},$$ (58)

with electric charge playing the role of mass. One major difference between the two laws is the sign of the force. The electrostatic force between two like charges is repulsive (i.e., $f>0$), whereas that between two unlike charges is attractive (i.e., $f<0$). On the other hand, the gravitational force between two masses is always attractive (since there is no such thing as a negative mass). Another major difference is the relative magnitude of the two forces. For instance, the electrostatic repulsion between two electrons is approximately $10^{42}$ times larger than the corresponding gravitational attraction.

The electrostatic force ${\bf f}_{ab}$ exerted by a charge $q_a$ on a second charge $q_b$, located a distance $r$ from the first charge, has the magnitude

$$f = \frac{q_a\,q_b}{4\pi\epsilon_0\,r^2},$$ (59)

and is directed radially away from the first charge if $f>0$, and radially towards it if $f<0$. The force ${\bf f}_{ba}$ exerted by the second charge on the first is equal and opposite to ${\bf f}_{ab}$, so that

$${\bf f}_{ba} = - {\bf f}_{ab},$$ (60)

in accordance with Newton's third law of motion.

Suppose that we have three point charges, $q_a$, $q_b$, and $q_c$. It turns out that electrostatic forces are superposable. That is, the force ${\bf f}_{ba}$ exerted by $q_b$ on $q_a$ is completely unaffected by the presence of $q_c$. Likewise, the force ${\bf f}_{ca}$ exerted by $q_c$ on $q_a$ is unaffected by the presence of $q_b$. Thus, the net force ${\bf f}_a$ acting on $q_a$ is the resultant of these two forces: i.e.,

$${\bf f}_a = {\bf f}_{ba} + {\bf f}_{ca}.$$ (61)

This rule can be generalized in a straightforward manner to the case where there are more than three point charges.