1.6 COULOMB'S LAW

Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges. Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges. Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by

$$F=k\frac{q_1{q}_2}{r^2}$$
(1.1)

How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges. However, the charges on the spheres were unknown, to begin with. How then could he discover a relation like Eq. (1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere is q. If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres. By symmetry, the charge on each sphere will be q/2**. Repeating this process, we can get charges q/2, q/4, etc. Coulomb varied the distance for a fixed pair of charges and measured the force for different separations. He then varied the charges in pairs, keeping the distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq. (1.1).

* A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.

** Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.

Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above. While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ ${10}^{-10}$ m).

Coulomb discovered his law without knowing the explicit magnitude of the charge. In fact, it is the other way round: Coulomb’s law can now be employed to furnish a definition for a unit of charge. In the relation, Eq. (1.1), k is so far arbitrary. We can choose any positive value of k. The choice of k determines the size of the unit of charge. In SI units, the value of k is about $9\times {10}^9$ $\frac{N{m}^2}{C^2}$.

The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4. Putting this value of k in Eq. (1.1), we see that for $q_1$ = $q_2$ = 1 C, r = 1 m $$F=9\ast {10}^{9}N$$

That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude $9\ast {10}^{9}N$. One coulomb is evidently too big a unit to be used. In practice, in electrostatics, one uses smaller units like 1 mC or 1 μC.

The constant k in Eq. (1.1) is usually put as $k=\frac{1}{4\pi {\epsilon }_0}$ for later convenience, so that Coulomb’s law is written as

$$F=\frac{1}{4\pi {\epsilon }_0}\frac{|q_1{q}_2|}{r^2}$$
(1.2)

${\epsilon }_0$ is called the permittivity of free space . The value of ${\epsilon }_0$ in SI units is $${\epsilon }_0=8.854\ast {10}^{-12}{C}^2{N}^{-1}{m}^{-2}$$

1.6

(a) Geometry and
(b) Forces between charges.

Since force is a vector, it is better to write Coulomb’s law in the vector notation. Let the position vectors of charges q₁ and q₂ be ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ respectively [see Fig.1.6(a)]. We denote force on q₁ due to q₂ by ${\mathbf{F}}_{12}$ and force on q2 due to q1 by ${\mathbf{F}}_{21}$. The two point charges q₁ and q₂ have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by ${\mathbf{r}}_{21}$: $${\mathbf{r}}_{21}={\mathbf{r}}_2-{\mathbf{r}}_1$$

In the same way, the vector leading from 2 to 1 is denoted by ${\mathbf{r}}_{12}$: $${\mathbf{r}}_{12}={\mathbf{r}}_1-{\mathbf{r}}_2$$

The magnitude of the vectors ${\mathbf{r}}_{21}$ and ${\mathbf{r}}_{12}$ is denoted by r₂₁ and r₁₂, respectively (r₁₂ = r₂₁). The direction of a vector is specified by a unit vector along the vector. To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: $${\hat{\mathbf{r}}}_{21}=\frac{{\hat{\mathbf{r}}}_{21}}{r_{21}},{\hat{\mathbf{r}}}_{12}=\frac{{\hat{\mathbf{r}}}_{12}}{r_{12}},{\hat{\mathbf{r}}}_{21}={\hat{\mathbf{r}}}_{12}$$

Coulomb’s force law between two point charges $q_1$ and $q_2$ located at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ is then expressed as

$${\mathbf{F}}_{21}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{21}^2}{\hat{\mathbf{r}}}_{21}$$
(1.3)

Some remarks on Eq. (1.3) are relevant:

• Equation (1.3) is valid for any sign of $q_1$ and $q_2$ whether positive or negative. If $q_1$ and $q_2$ are of the same sign (either both positive or both negative), $F_{21}$ is along $r_{21}$, which denotes repulsion, as it should be for like charges. If $q_1$ and $q_2$ are of opposite signs, $F_{21}$ is along – $r_{21}$(= $r_{12}$), which denotes attraction, as expected for unlike charges. Thus, we do not have to write separate equations for the cases of like and unlike charges. Equation (1.3) takes care of both cases correctly [Fig. 1.6(b)].
• The force $F_{12}$ on charge $q_1$ due to charge $q_2$, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e., $$F_{12}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{21}^2}{r}_{12}=-F_{21}$$ Thus, Coulomb’s law agrees with the Newton’s third law.
• Coulomb’s law [Eq. (1.3)] gives the force between two charges $q_1$ and $q_2$ in vacuum. If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter. We shall consider electrostatics in matter in the next chapter.

Example 1.5

A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in Fig. 1.7(a). The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen). Spheres A and B are touched by uncharged spheres C and D respectively, as shown in Fig. 1.7(b). C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in Fig. 1.7(c). What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes. Ignore the sizes of A and B in comparison to the separation between their centres.

1.7

VIEW SOLUTION

Let the original charge on sphere A be q and that on B be q'. At a distance r between their centres, the magnitude of the electrostatic force on each is given by $$F=\frac{1}{4\pi {\epsilon }_0}\frac{q{q}^{\prime }}{r^2}$$ neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q'/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is $$F=\frac{1}{4\pi {\epsilon }_0}\frac{(q/2)(q^{\prime }/2)}{(r/2{)}^2}=\frac{1}{4\pi {\epsilon }_0}\frac{q{q}^{\prime }}{r^2}=F$$ Thus the electrostatic force on A, due to B, remains unaltered.