Question

For n charges \[q_1, q_2, \ldots, q_n\] at positions \[r_1, r_2, \ldots, r_n\], where \[r_{ij}\] is the distance between charge i and charge j, the total electrostatic potential energy of the system is:

Options

• \[U = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n \sum_{j>i}^n \frac{q_i q_j}{r_{ij}}\]

• \[U = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n \frac{q_i}{r_i}\]

• \[U = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n \frac{q_i^2}{r_{ii}}\]

• \[U = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n \sum_{j>i}^n \frac{q_i q_j}{r_{ij}^2}\]

Answer 1

For an n-charge system, \[U = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n \sum_{j>i}^n \frac{q_i q_j}{r_{ij}}\], where \[r_{ij}\] is the distance between charge i and charge j. The condition j > i ensures each unique pair contributes exactly one term.