Question

Obtain an expression for potential energy of a system of N point charges.

Answer 1

The equation,

`U = 1/(4piε_0) . (q_1q_2)/r_12` ves an expression or potential energy for a system of two charges.

We now examine the situation for an N-point charge system.

In bringing a charge q3 from ∞ to `C (vec(r_3))` work has to be done against electrostatic forces of both q₁ and q₂.

∴ W₃ = ((Potential at C due to q₁ and q₂) x q₃

`1/(4piε_0) [q_1/r_13 + q_2/r_23] xx q_3`

∴ `W_3 = 1/(4piε_0) [(q_1q_3)/r_13 + (q_2q_3)/r_23]`

Similarly, in bringing charge q4 from ∞ to `D(vecr_4)`work has done against forces of q₁, q₂ and q₃ 

∴ `W_4 = 1/(4piε_0)[(q_1q_4)/r_14 + (q_2q_4)/r_24 + (q_3q_4)/r_34]`

Similarly, we can express the electrostatic potential energy of an N-body system.

point charges at `vec(r_1), vec(r_2),.... vec(r_N)  "as"`

`U = 1/(4piε_0)sum_"all pairs" (q_jq_k)/(r_jk)`