Physics · Electrostatic Potential

Two charges \( q_1 \) and \( q_2 \) are placed at position vectors \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) in an external electric field with potential \( V(\mathbf{r}) \). If \( r_{12} \) is the separation between them, the total electrostatic potential energy of the system is:

1. \( q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) \)
2. \( q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{q_1 q_2}{4\pi\varepsilon_0 r_{12}} \)
3. \( \frac{q_1 q_2}{4\pi\varepsilon_0 r_{12}} \)
4. \( (q_1 + q_2)[V(\mathbf{r}_1) - V(\mathbf{r}_2)] + \frac{q_1 q_2}{4\pi\varepsilon_0 r_{12}^2} \)