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To calculate the electric potential energy of the system in this square configuration, we need to consider the interactions between each pair of charges. Given that the side length of the square is \( a \), let's break down the approach to solve this:

### Step-by-Step Solution:
1. **Identify Charges and Distances**:
   - The charges are positioned at the vertices of the square. They are: \( q \), \( 2q \), \( q \), and \( -3q \).
   - Each side of the square has a length of \( a \), so adjacent charges are separated by \( a \).
   - Diagonal charges are separated by a distance of \( \sqrt{2}a \).

2. **Calculate Pairwise Potential Energies**:
   For two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \), the potential energy \( U \) is given by:
   \[
   U = \frac{k \cdot q_1 \cdot q_2}{r}
   \]
   where \( k \) is Coulomb's constant.

3. **Evaluate All Interactions**:
   - Between each adjacent pair (distance \( a \)):  
     - \( U_{q \text{ and } 2q} = \frac{k \cdot q \cdot 2q}{a} = \frac{2kq^2}{a} \)
     - \( U_{q \text{ and } q} = \frac{k \cdot q \cdot q}{a} = \frac{kq^2}{a} \)
     - \( U_{2q \text{ and } q} = \frac{k \cdot 2q \cdot q}{a} = \frac{2kq^2}{a} \)
     - \( U_{q \text{ and } -3q} = \frac{k \cdot q \cdot (-3q)}{a} = -\frac{3kq^2}{a} \)

   - Between each diagonal pair (distance \( \sqrt{2}a \)):  
     - \( U_{q \text{ and } q} = \frac{k \cdot q \cdot q}{\sqrt{2}a} = \frac{kq^2}{\sqrt{2}a} \)
     - \( U_{2q \text{ and } -3q} = \frac{k \cdot 2q \cdot (-3q)}{\sqrt{2}a} = -\frac{6kq^2}{\sqrt{2}a} \)

4. **Sum the Energies**:
   Add all these pairwise interactions to get the total electric potential energy of the system.

Would you like me to continue with the calculations, or is there any specific part you need further clarification on?

### Related Questions:
1. How does changing one of the charges affect the potential energy?
2. What happens to the potential energy if the square's side length doubles?
3. Can this configuration result in a stable equilibrium?
4. How would the potential energy change if all charges were positive?
5. How is electric potential energy different from electric potential?

### Tip:
Always keep track of both the magnitude and sign of each charge in potential energy calculations, as they affect attraction or repulsion in the system.

محاسبه انرژی پتانسیل الکتریکی شکل زیر. طول ضلع مربع a است.

Calculate the electric potential energy of a system of charges arranged in a square with side length 'a'. In this configuration, charges are positioned at each corner with magnitudes q, 2q, -3q, and q. This problem uses Coulomb's law and involves calculating pairwise interactions between charges.

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