Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller...
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 | | q1 | q2 | | |
| | | ||||
| | | | |||
| | | r 2 | | | |
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 Charles Augustin de different pairs of charges at different distances, Coulomb arrived at the relation, Eq. (1.1).
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 × 109. 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 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 × 109. 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
q1 = q2 = 1 C, r = 1 m F = 9 × 109 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 × 109 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 = 1/4πε0 for later convenience, so that Coulomb’s law is written as
| 1 | | | q q | | | | ||||
| F = | | | | 1 | 2 | | | |||
| 4 π ε0 | | | r 2 | | | |||||
ε0 is called the permittivity of free space . The value of ε0 in SI units is
ε 0 = 8.854 × 10–12 C2 N–1m–2
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 r1 and r2 respectively [see Fig.1.6(a)]. We denote force on
q1 due to q2 by F12 and force on q2 due to q1 by
F21. 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 r21:: r21 = r2 – r1
In the same way, the vector leading from 2 to
1 is denoted by
r12:
r12 = r1 – r2 = – r21
The magnitude of the vectors r21 and r12 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:
| | r21 | | | r12 | , | | | |
| r 21 = | | , | r 12 = | | r 21 = | r 12 | ||
| r21 | r12 | |||||||
| | | | | | |
Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as
| 1 | | | q1 | q2 | |
| 4 π ε | | | r 2 | r 21 | |
| | o | | | 21 | |
Some remarks on Eq. (1.3) are relevant:
• Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative. If q1 and q2 are of the same sign (either both positive or both negative), F21 is along rˆ 21, which denotes repulsion, as it should be for like charges. If q1 and q2 are of opposite signs, F21 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 F12 on charge q1 due to charge q2, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e.,
| | | 1 | | | q1 | q2 | | | |
| F12 | = | 4 Ï€ ε | 0 | | r 2 | r | 12 | = −F21 | |
| | | | | | 12 | | | | |
Thus, Coulomb’s law agrees with the Newton’s third law.
• Coulomb’s law [Eq. (1.3)] gives the force between two charges q1 and q2 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.
