Three identical charges are placed at corners of a equilateral triangel of side l. If force between any two charges is F, the work required to double the dimensiions of triangle is

Options

A. `-3Fl`
B. `3Fl`
C. `(-3//2)Fl`
D. `(3//2)Fl`

Explanation

To find the work required to double the dimensions of an equilateral triangle formed by three identical charges, we can follow these steps:

### Step 1: Understand the Initial Configuration
We have three identical charges \( Q \) placed at the corners of an equilateral triangle with side length \( L \). The force between any two charges is given as \( F \).

### Step 2: Calculate the Initial Potential Energy
The potential energy \( U \) between two point charges is given by the formula:
\[ U = k \frac{Q_1 Q_2}{r} \]
For three identical charges at the corners of the triangle, the potential energy due to each pair of charges is:
\[ U_{pair} = k \frac{Q^2}{L} \]
Since there are three pairs of charges (between each pair of corners), the total initial potential energy \( U_i \) is:
\[ U_i = 3 \cdot k \frac{Q^2}{L} = \frac{3kQ^2}{L} \]

### Step 3: Calculate the Final Potential Energy
When the dimensions of the triangle are doubled, the new side length becomes \( 2L \). The potential energy for each pair of charges in the new configuration is:
\[ U'_{pair} = k \frac{Q^2}{2L} \]
The total final potential energy \( U_f \) is:
\[ U_f = 3 \cdot k \frac{Q^2}{2L} = \frac{3kQ^2}{2L} \]

### Step 4: Calculate the Work Done
The work done \( W \) by an external agent to change the configuration is equal to the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values we calculated:
\[ W = \frac{3kQ^2}{2L} - \frac{3kQ^2}{L} \]
To simplify, we can factor out \( \frac{3kQ^2}{L} \):
\[ W = \frac{3kQ^2}{L} \left( \frac{1}{2} - 1 \right) = \frac{3kQ^2}{L} \left( -\frac{1}{2} \right) = -\frac{3kQ^2}{2L} \]

### Step 5: Relate to the Given Force \( F \)
From the problem, we know that the force \( F \) between any two charges is given by:
\[ F = k \frac{Q^2}{L^2} \]
We can express \( kQ^2/L \) in terms of \( F \):
\[ k \frac{Q^2}{L} = F \cdot L \]
Substituting this into the work done equation:
\[ W = -\frac{3}{2} (F \cdot L) \]

### Final Answer
Thus, the work required to double the dimensions of the triangle is:
\[ W = -\frac{3}{2} FL \]