Superposition Principle

The principle of superposition is to calculate the electric force experienced by a charge q1 due to other charges q2, q3 ……. qn. 

The total force on a given charge is the vector sum of the forces exerted on it due to all other charges.

The force on q1 due to q2 

Vector F12 = k q1 q2 / (r21)^2     (r21 cap)

similarly,  force on q1 due to q3 

Vector F21 = k q1 q2 / (r31)^2     (r31 cap)

The total force F1 on the charge q1 by all other charges is, 

Vector  F1 = Vector [F12+F13+F14+.........+F1n]

So 

Vector F1=1/4πεo[q1q2/r21^2 (r21cap) + q1q3/r31^2  (r31cap) + ..... + q1q n/r n1^2   (r n1 cap )]

Note:

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.

Coulomb's Law in vector form

If vector F21 is the force exerted on charge q2 by charge q1 (Fig.),

Vector  F21 = k q1 q2 / (r12)^2     (r12 cap)

where unit vector r 12 cap is the unit vector  from q1 to q2.

If vector F12 is the force exerted on q1 due to q2,

Vector  F12 = k q1 q2 / (r21)^2     (r21 cap)

where unit vector r 21 cap is the unit vector  from q2 to q1.                           

                        Fig. Coulomb’s law in vector form                               

[Both unit vector u r 21 and r 12 have the same magnitude, and are oppositely directed]

      Vector F12 = k q1 q2 / (r12)^2     (-r12 cap)

or   Vector F12 = - k q1 q2 / (r12)^2     (r12 cap)

or   Vector F12 = -Vector F21

So, the forces exerted by charges on each other are equal in magnitude and opposite in direction.

Note :

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.

Coulomb's Law

Coulomb’s law : Coulomb’s law states that the force of attraction or repulsion between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The direction of forces is along the line joining the two point charges.

Let q1 and q2 be two point charges placed in air or vacuum at a distance r apart (Fig.) 

              

Then, according to Coulomb’s law, 

F α q1 q2 / r^2

F= k q1 q2 / r^2

k is a constant of proportionality 

In air or vacuum, k = 1/4πεo, where εo is the permittivity of free space (i.e., vacuum) and the value of εo is 8.854 × 10^−12 C2 N^−1 m^−2.

F = q1 q2 / 4πεor^2      ...(1)

and 1/4πεo = 9 * 10^9 N m^2 C^-2

In the above equation, if q1 = q2 = 1C and  r = 1m then,

F = (9 * 10^9) x (1*1/1^2) = 9 * 10^9 N

One Coulomb is defined as the quantity of charge, which when placed at a distance of 1 metre in air or vacuum from an equal and similar charge, experiences a repulsive force of 9 × 10^9 N.

If the charges are situated in a medium of permittivity ε, then the magnitude of the force between them will be,

Fm = q1 q2 / 4πε r^2      ...(2)

Dividing equation (1) by (2)

F / Fm =  ε/εo = εr

The ratio  ε/εο = εr, is called the relative permittivity or dielectric constant of the medium. The value of εr for air or vacuum is 1.

∴  ε = εo εr

Since Fm =F/εr , the force between two point charges depends on the nature of the medium in which the two charges are situated.

Note :

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.