Question

What is meant by the electric potential energy of the system of point charges? Compute the total electric potential energy of the system of charges given in the figure.
\includegraphics[width=0.5\linewidth]{image6.png}

Hint:

The total electric potential energy of a system of point charges is calculated by summing the potential energies of all pairs of charges, where each pair's potential energy is given by \( \frac{k_e q_i q_j}{r_{ij}} \). For symmetric charge configurations, the distances can be easily calculated using geometry.

Answer 1

The electric potential energy of a system of point charges is the work done to assemble the system of charges from infinity. It is given by the formula:
\[ U = \sum_{i<j} \frac{k_e q_i q_j}{r_{ij}}, \] where:
- \( U \) is the total electric potential energy of the system,
- \( k_e = 9 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \) is Coulomb's constant,
- \( q_i \) and \( q_j \) are the charges,
- \( r_{ij} \) is the distance between charges \( i \) and \( j \).
In the given figure, we have a system of four point charges arranged at the corners of a square. The charges are:
- \( +q \) at \( (0,0) \),
- \( -q \) at \( (a,0) \),
- \( -q \) at \( (a,a) \),
- \( +q \) at \( (0,a) \).
The distances between each pair of charges are either \( a \) (side of the square) or \( \sqrt{2}a \) (diagonal of the square).
Step 1: Calculate the potential energy due to the interactions between charges on adjacent sides (distance \( a \)):
- Interaction between \( +q \) and \( -q \) at \( (0,0) \) and \( (a,0) \): \[ U_1 = \frac{k_e (+q)(-q)}{a} = -\frac{k_e q^2}{a}. \] - Interaction between \( -q \) and \( +q \) at \( (a,0) \) and \( (a,a) \): \[ U_2 = \frac{k_e (-q)(+q)}{a} = -\frac{k_e q^2}{a}. \] - Interaction between \( +q \) and \( -q \) at \( (a,a) \) and \( (0,a) \): \[ U_3 = \frac{k_e (+q)(-q)}{a} = -\frac{k_e q^2}{a}. \] - Interaction between \( -q \) and \( +q \) at \( (0,a) \) and \( (0,0) \): \[ U_4 = \frac{k_e (-q)(+q)}{a} = -\frac{k_e q^2}{a}. \] Thus, the total potential energy due to adjacent charges is: \[ U_{\text{adj}} = 4 \left( -\frac{k_e q^2}{a} \right) = -\frac{4 k_e q^2}{a}. \] Step 2: Calculate the potential energy due to the interactions between charges on the diagonals (distance \( \sqrt{2}a \)):
- Interaction between \( +q \) at \( (0,0) \) and \( -q \) at \( (a,a) \): \[ U_5 = \frac{k_e (+q)(-q)}{\sqrt{2}a} = -\frac{k_e q^2}{\sqrt{2}a}. \] - Interaction between \( -q \) at \( (a,0) \) and \( +q \) at \( (0,a) \): \[ U_6 = \frac{k_e (-q)(+q)}{\sqrt{2}a} = -\frac{k_e q^2}{\sqrt{2}a}. \] Thus, the total potential energy due to diagonal charges is: \[ U_{\text{diag}} = 2 \left( -\frac{k_e q^2}{\sqrt{2}a} \right) = -\frac{2 k_e q^2}{\sqrt{2}a}. \] Step 3: Total electric potential energy of the system: Now, the total electric potential energy of the system is the sum of the contributions from adjacent and diagonal interactions: \[ U_{\text{total}} = U_{\text{adj}} + U_{\text{diag}} = -\frac{4 k_e q^2}{a} - \frac{2 k_e q^2}{\sqrt{2}a}. \] Thus, the total electric potential energy of the system is: \[ U_{\text{total}} = -\frac{4 k_e q^2}{a} - \frac{2 k_e q^2}{\sqrt{2}a}. \] This is the total electric potential energy of the system of point charges arranged in a square.