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 will analyze the forces acting between the charges at the corners of the square.

### Step 1: Understand the Configuration
We have three equal charges \( q_1 \), \( q_2 \), and \( q_3 \) placed at three corners of a square. Let's denote the side length of the square as \( a \).

### Step 2: Calculate the Force \( F_{12} \)
The force between \( q_1 \) and \( q_2 \) (which are at adjacent corners) can be calculated using Coulomb's Law:

\[ F_{12} = k \frac{q_1 q_2}{r^2} \]

Here, \( r \) is the distance between \( q_1 \) and \( q_2 \), which is equal to \( a \) (the side of the square). Since the charges are equal, we can denote them as \( q \):

\[ F_{12} = k \frac{q^2}{a^2} \]

### Step 3: Calculate the Force \( F_{13} \)
The force between \( q_1 \) and \( q_3 \) (which are at opposite corners of the square) is also calculated using Coulomb's Law. The distance between \( q_1 \) and \( q_3 \) is the diagonal of the square, which can be calculated using the Pythagorean theorem:

\[ r = \sqrt{a^2 + a^2} = a\sqrt{2} \]

Thus, the force \( F_{13} \) is given by:

\[ F_{13} = k \frac{q_1 q_3}{(a\sqrt{2})^2} = k \frac{q^2}{2a^2} \]

### Step 4: Find the Ratio \( \frac{F_{12}}{F_{13}} \)
Now we can find the ratio of the magnitudes of the forces:

\[ \frac{F_{12}}{F_{13}} = \frac{k \frac{q^2}{a^2}}{k \frac{q^2}{2a^2}} = \frac{1}{\frac{1}{2}} = 2 \]

### Conclusion
The ratio of the magnitudes of the forces \( F_{12} \) and \( F_{13} \) is:

\[ \frac{F_{12}}{F_{13}} = 2 \]

### Final Answer
Thus, the answer is \( 2 \).

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