Question

Derive an expression for potential energy of a system of two point charges.

Answer 1

1. Electrostatic potential energy of a system of point charges is defined as the total amount of work done to assemble the system of charges by bringing them from infinity to their present locations.
   
2. Let us consider 2 charges q₁ and q₂ with position vectors r₁ and r₂ relative to the origin.
3. We consider the charges q₁ and q₂ initially at infinity and determine the work done by an external agency to bring the charges to the given locations.

4. In bringing the first charge q₁ to position A(`barr_1`) no work is done because there is no external field against which work is no work is done because there is no external field against which work is required to be done as charge q₂ is still at infinity i.e., W₁ = 0. This charge produces a potential in space given by `V_l=1/(4piepsi_0)q_1/r_1`
   where r₁ is the distance of point A from the origin.

5. When we bring charge q₂ from infinity to B(`barr_2`) at a distance r₁₂, from q₁, work done is `W_2=("potential at  B due to charge"  q_1)xxq_2``=q_1/(4piepsi_0r_12)xxq_2,("where" AB = r_12)`

6. This work done in bringing the two charges to their respective locations is stored as the potential energy of the configuration of two charges.
   `U=1/(4piepsi_0)(q_1q_2)/r_12`