CBSE NCERT Class 12 : 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(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.) 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. 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.

      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, 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. Putting this value of k, we see that for
q1 = q2 = 1 C, r = 1 m
F = 9 × 10^9 N
      That is, 1C is the charge that when placed at a distance of 1m from another charge of the same
magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 10^9 N. One coulomb is evidently too big a unit to be used. In practice, in electrostatics, one uses smaller units like 1mC or 1μC.
  The constant k is usually put as     for later convenience, so that Coulomb’s law is written as

 

is known as permitivity in free space, its value in SI unit is,8.854 × 10^(–12) C^2 N^(–1)m^(–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. 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:

 

Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as

 

 Some remarks on Eq. above are relevant:
• Equation 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 ˆr 21, which denotes repulsion, as it should be for like charges. If q1 and q2 are of opposite signs, F21 is along –ˆr 21(=ˆr 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 takes care of both cases correctly.
• The force F12 on charge q1 due to charge q2, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e.,

Thus, Coulomb’s law agrees with the Newton’s third law.
• Coulomb’s law 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.

 FORCES BETWEEN MULTIPLE CHARGES

The mutual electric force between two charges is given by Coulomb’s law. How to calculate the force on a charge where there are not one but several charges around? Consider a system of n stationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question. Recall that forces of mechanical origin add according to the parallelogram law of addition. Is the same true for forces of electrostatic origin? Experimentally it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges. This is termed as the principle of superposition. To better understand the concept, consider a system of three charges q1, q2 and q3. The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges. Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by even though other charges are present.

In the same way, the force on q1 due to q3, denoted by F13, is given by

which again is the Coulomb force on q1 due to q3, even though other charge q2 is present.
        Thus the total force F1 on q1 due to the two charges q2 and q3 is given as

      The above calculation of force can be generalised to a system of charges more than three. The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn. The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n:

The vector sum is obtained as usual by the parallelogram law of addition of vectors.All of electrostats is basically a consequence of Coulomb’s law and the superposition principle.