The work which is required to be done to make an arrangement of four paritcles each having a charge `+q` such that the particles lie at the four corners of a square of side a is

Options

A. `(4 + sqrt(2)) (kq^(2))/(a)`
B. `4 (kq^(2))/(a )`
C. `( 2 + sqrt(2)) (kq^(2))/(a) `
D. `2 (kq^(2))/(a )`

Answer

To find the work required to arrange four particles, each having a charge \( +q \), at the corners of a square of side \( a \), we can follow these steps:

### Step 1: Understand the Initial Condition
Initially, the four charges are at infinity, where the potential energy is zero. Therefore, the initial potential energy \( U_i \) is:
\[ U_i = 0 \]

### Step 2: Calculate the Potential Energy when Charges are Arranged
When the charges are brought to the corners of the square, we need to calculate the potential energy of the system. The potential energy \( U_f \) of a system of point charges is given by the sum of the potential energies of all pairs of charges.

### Step 3: Identify the Pairs of Charges
In a square arrangement of four charges, we have:
- 4 pairs of adjacent charges (1-2, 2-3, 3-4, 4-1)
- 2 pairs of diagonal charges (1-3, 2-4)

### Step 4: Calculate the Potential Energy of Adjacent Pairs
The potential energy \( U \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ U = k \frac{q_1 q_2}{r} \]
For adjacent pairs (distance \( a \)):
\[ U_{\text{adjacent}} = k \frac{q \cdot q}{a} = k \frac{q^2}{a} \]
Since there are 4 adjacent pairs:
\[ U_{\text{adjacent total}} = 4 \cdot k \frac{q^2}{a} = \frac{4kq^2}{a} \]

### Step 5: Calculate the Potential Energy of Diagonal Pairs
For diagonal pairs (distance \( \sqrt{2}a \)):
\[ U_{\text{diagonal}} = k \frac{q \cdot q}{\sqrt{2}a} = k \frac{q^2}{\sqrt{2}a} \]
Since there are 2 diagonal pairs:
\[ U_{\text{diagonal total}} = 2 \cdot k \frac{q^2}{\sqrt{2}a} = \frac{2kq^2}{\sqrt{2}a} \]

### Step 6: Combine the Potential Energies
Now, we can find the total potential energy \( U_f \) when the charges are arranged:
\[ U_f = U_{\text{adjacent total}} + U_{\text{diagonal total}} = \frac{4kq^2}{a} + \frac{2kq^2}{\sqrt{2}a} \]

### Step 7: Simplify the Expression
To combine the terms, we can factor out \( kq^2/a \):
\[ U_f = k \frac{q^2}{a} \left( 4 + \frac{2}{\sqrt{2}} \right) \]
Since \( \frac{2}{\sqrt{2}} = \sqrt{2} \):
\[ U_f = k \frac{q^2}{a} \left( 4 + \sqrt{2} \right) \]

### Step 8: Calculate the Work Done
The work done \( W \) to arrange the charges is equal to the change in potential energy:
\[ W = U_f - U_i = U_f - 0 = U_f \]
Thus, the work required is:
\[ W = k \frac{q^2}{a} \left( 4 + \sqrt{2} \right) \]

### Final Answer
The work required to arrange the four charges at the corners of a square of side \( a \) is:
\[ W = k \frac{q^2}{a} \left( 4 + \sqrt{2} \right) \]