Three equal charges are placed on the three corners of a square. If the force between `q_(1)` and `q_(2)` is `F_(12)` and that between `q_(1)` and `q_(3)` is `F_(13)`, then the ratio of magnitudes `(F_(12)//F_(13))` is

Options

A. `1//2`
B. 2
C. `1//sqrt2`
D. `sqrt2`

Answer

To solve the problem, we need to find the ratio of the magnitudes of the forces \( F_{12} \) and \( F_{13} \) between three equal charges \( q_1 \), \( q_2 \), and \( q_3 \) placed at the corners of a square.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have three equal charges \( q \) placed at the corners of a square. Let's denote the side length of the square as \( A \).
- The charges are positioned as follows:
- \( q_1 \) at (0, 0)
- \( q_2 \) at (A, 0)
- \( q_3 \) at (0, A)

2. **Calculating the Force \( F_{12} \)**:
- The force \( F_{12} \) is the electrostatic force between charges \( q_1 \) and \( q_2 \).
- The distance between \( q_1 \) and \( q_2 \) is \( A \).
- Using Coulomb's law, the force is given by:
\[ F_{12} = \frac{k \cdot q_1 \cdot q_2}{A^2} \]
- Since \( q_1 = q_2 = q \), we have:
\[ F_{12} = \frac{k \cdot q^2}{A^2} \]

3. **Calculating the Force \( F_{13} \)**:
- The force \( F_{13} \) is the electrostatic force between charges \( q_1 \) and \( q_3 \).
- The distance between \( q_1 \) and \( q_3 \) is the diagonal of the square, which can be calculated using the Pythagorean theorem:
\[ \text{Distance} = \sqrt{A^2 + A^2} = A\sqrt{2} \]
- Using Coulomb's law, the force is given by:
\[ F_{13} = \frac{k \cdot q_1 \cdot q_3}{(A\sqrt{2})^2} \]
- Since \( q_1 = q_3 = q \), we have:
\[ F_{13} = \frac{k \cdot q^2}{2A^2} \]

4. **Finding the Ratio \( \frac{F_{12}}{F_{13}} \)**:
- Now we can find the ratio of the forces:
\[ \frac{F_{12}}{F_{13}} = \frac{\frac{k \cdot q^2}{A^2}}{\frac{k \cdot q^2}{2A^2}} = \frac{1}{\frac{1}{2}} = 2 \]

5. **Final Result**:
- Therefore, the ratio of the magnitudes of the forces is:
\[ \frac{F_{12}}{F_{13}} = 2 \]