Three identical charges are placed on three vertices of a square. If the force acting between `q_1 and q_2` is `F_(12)` and between `q_1 and q_3` is `F_13` then `(F_13)/(F_12) =` ________.

Options

A. `1/2`
B. `2`
C. `1/(sqrt2)`
D. `sqrt(2)`

Answer

To solve the problem, we need to find the ratio of the forces \( F_{13} \) (the force between charges \( q_1 \) and \( q_3 \)) and \( F_{12} \) (the force between charges \( q_1 \) and \( q_2 \)) when the charges are placed at the vertices of a square.

### Step-by-Step Solution:

1. **Identify the Charges and Distances**:
- Let the charges at the vertices of the square be \( q_1 \), \( q_2 \), and \( q_3 \).
- The distance between \( q_1 \) and \( q_2 \) (which are adjacent vertices) is \( A \) (the side length of the square).
- The distance between \( q_1 \) and \( q_3 \) (which are opposite vertices) is \( A\sqrt{2} \) (the diagonal of the square).

2. **Apply Coulomb's Law**:
- According to Coulomb's law, the force between two point charges is given by:
\[ F = k \frac{q_1 q_2}{r^2} \]
- For \( F_{12} \) (force between \( q_1 \) and \( q_2 \)):
\[ F_{12} = k \frac{q^2}{A^2} \]
- For \( F_{13} \) (force between \( q_1 \) and \( q_3 \)):
\[ F_{13} = k \frac{q^2}{(A\sqrt{2})^2} = k \frac{q^2}{2A^2} \]

3. **Calculate the Ratio \( \frac{F_{13}}{F_{12}} \)**:
- Now, we can find the ratio of the two forces:
\[ \frac{F_{13}}{F_{12}} = \frac{k \frac{q^2}{2A^2}}{k \frac{q^2}{A^2}} = \frac{1/2}{1} = \frac{1}{2} \]

4. **Final Answer**:
- Therefore, the ratio \( \frac{F_{13}}{F_{12}} \) is:
\[ \frac{F_{13}}{F_{12}} = \frac{1}{2} \]