Question

Obtain an expression for potential energy due to a collection of three point charges which are separated by finite distances.

Answer 1

Electrostatic potential energy for a collection of point charges:

The electric potential at a point at a distance r from point charge ql is given by

V = `1/(4piε_0) "q"_1/"r"`  ....(1)

This potential V is the work done to bring a unit positive charge from infinity to the point. Now if the charge q₂ is brought from infinity to that point at a distance r from qp the work done is the product of q₂ and the electric potential at that point. Thus we have W = q₂V …… (2)
This work done is stored as the electrostatic potential energy U of a system of charges q₁ and q₂ separated by a distance r. Thus we have

U = `"q"_2"V" = 1/(4piε_0) ("q"_1"q"_2)/"r"`   ....(3)

The electrostatic potential energy depends only on the distance between the two point charges. In fact, the expression (3) is derived by assuming that q₁ is fixed and q₂ is brought from infinity. The equation (3) holds true when q₂ is fixed and q₁ is brought from infinity or both q₂and q₂ are simultaneously brought from infinity to a distance r between them.
Three charges are arranged in the following configuration as shown in Figure.

Electrostatic potential energy for collection of point charges

To calculate the total electrostatic potential energy, we use the following procedure. We bring all the charges one by one and arrange them according to the configuration.

1. Bringing a charge q₁ from infinity to the point A requires no work, because there are no other charges already present in the vicinity of charge q₁

2. To bring the second charge q₂ to the point B, work must be done against the electric field created by the charge q₁ So the work done on the charge q₁ is W = q₂V_1B. Here V_1B is the electrostatic potential due to the charge q₁ at point B.
   U = `1/(4 pi ε_0) ("q"_1"q"_2)/"r"_12`   .....(4)
   

   Note that the expression is same when q₂ is brought first and then q₁ later.

3. Similarly to bring the charge q3 to the point C, work has to be done against the total electric field due to both charges q₁ and q₂. So the work done to bring the charge q₃ is = q₃ (V_1C + V_2C). Here V_1C is the electrostatic potential due to charge q₁ at point C and V_2C is the electrostatic potential due to charge q₂ at point C. The electrostatic potential is
   U = `1/(4 pi ε_0) (("q"_1"q"_3)/"r"_13 + ("q"_2"q"3)/"r"_23)`    ....(5)
4. Adding equations (4) and (5), the total electrostatic potential energy for the system of three charges q₁,q₂ and q₃ is
   U = `1/(4 pi ε_0) (("q"_1"q"_2)/"r"_12 + ("q"_1"q"3)/"r"_13 + ("q"_2"q"_3)/"r"_23)`    ....(6)
   

   Note that this stored potential energy U is equal to the total external work done to assemble the three charges at the given locations. The expression (6) is same if the charges are brought to their positions in any other order. Since the Coulomb force is a conservative force, the electrostatic potential energy is independent of the manner in which the configuration of charges is arrived at.