Three equal charges `q_1 ,q_2 ,q_3` are placed at the three corners ABC of a square ABCD. If the force between the charges at A and B (on `q_1 and q_2`) is `F_(12)` and that between A and C is `F_(13)` then the ratio of magnitudes `F_(12)` and `F_(13)` is

Options

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

Answer

To solve the problem of finding the ratio of the magnitudes of the forces \( F_{12} \) and \( F_{13} \) between three equal charges placed at the corners of a square, we can follow these steps:

### Step 1: Understand the Configuration
We have three equal charges \( q_1, q_2, \) and \( q_3 \) placed at the corners A, B, and C of a square ABCD. The distances between the charges are determined by the geometry of the square.

### Step 2: Identify the Distances
Let the side length of the square be \( a \).
- The distance between charges at A and B (i.e., \( F_{12} \)) is equal to the side of the square: \( d_{AB} = a \).
- The distance between charges at A and C (i.e., \( F_{13} \)) is the diagonal of the square: \( d_{AC} = \sqrt{2}a \).

### Step 3: Calculate the Forces
Using Coulomb's Law, the force between two point charges is given by:
\[ F = k \frac{q_1 q_2}{r^2} \]
where \( k \) is Coulomb's constant.

#### Calculate \( F_{12} \):
\[ F_{12} = k \frac{q_1 q_2}{d_{AB}^2} = k \frac{q^2}{a^2} \]

#### Calculate \( F_{13} \):
\[ F_{13} = k \frac{q_1 q_3}{d_{AC}^2} = k \frac{q^2}{(\sqrt{2}a)^2} = k \frac{q^2}{2a^2} \]

### Step 4: Find the Ratio \( \frac{F_{12}}{F_{13}} \)
Now we can find the ratio of the magnitudes of the forces:
\[ \frac{F_{12}}{F_{13}} = \frac{k \frac{q^2}{a^2}}{k \frac{q^2}{2a^2}} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
Thus, the ratio of the magnitudes \( F_{12} \) and \( F_{13} \) is:
\[ \frac{F_{12}}{F_{13}} = 2 \]