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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
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 Equation.
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, equation.
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 ().
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, 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 . The unit of charge that results from this choice is called a coulomb. Putting this value of k, we see that for
q1 = q2 = 1 C,
r = 1 m
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 . 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 is usually put as for later convenience, so that Coulomb’s law is written as
is called the permittivity of free space. The value of in SI units is latex F_{21} = \frac{1}{4 \pi \epsilon_o } \frac{q_1 q_2}{r^{2}_{21}} latex F_{21} = F_{12} latex F_{12} = \frac{1}{4 \pi \epsilon_o } \frac{q_1 q_2}{r^{2}_{12}} latex F_{13} = \frac{1}{4 \pi \epsilon_o } \frac{q_1 q_2}{r^{2}_{13}} latex F_1 = F_{12} + F_{13} latex F_1 = F_{12} + F_{13} + F_{14} + ….. $
The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.