Question

Find out total electric potential energy of the system of charges, shown in the figure:

Hint:

The potential energy of a system of charges depends on the pairwise interactions between all charges, and it is calculated using Coulomb's law.

Answer 1

Step 1: Formula for Potential Energy

The total electric potential energy \( U \) of a system of charges is given by the formula:

\[ U = \sum_{iwhere \( k \) is Coulomb's constant, \( q_i \) and \( q_j \) are the charges, and \( r_{ij} \) is the distance between the charges.

Step 2: Calculate Potential Energy for the Three Charges

For the given system, the charges are \( +2q \), \( -q \), and \( -q \). The distance between any two charges is \( a \). Thus, the potential energy between each pair of charges is:

• Between \( +2q \) and \( -q \): \[ U_{12} = \frac{k \times 2q \times (-q)}{a} = -\frac{2kq^2}{a} \]
• Between \( -q \) and \( -q \): \[ U_{23} = \frac{k \times (-q) \times (-q)}{a} = \frac{kq^2}{a} \]
• Between \( +2q \) and \( -q \) (again): \[ U_{13} = \frac{k \times 2q \times (-q)}{a} = -\frac{2kq^2}{a} \]

Step 3: Total Potential Energy

The total potential energy is the sum of all pairwise potential energies:

\[ U_{\text{total}} = U_{12} + U_{23} + U_{13} = -\frac{2kq^2}{a} + \frac{kq^2}{a} - \frac{2kq^2}{a} = -\frac{3kq^2}{a} \]

Step 4: Conclusion

Thus, the total electric potential energy of the system is \( U = -\frac{3kq^2}{a} \).