Two identical short bar magnets at a distance `d_(1) = 2 m ` apart are placed with their magnetic moment parallel to each other . Now keeping same direction of magnetic moment they are placed at `d_(2) = 3` m apart . By which factor force between them changes ?

Options

A. `(F_(1))/(F_(2)) = (81)/(16)`
B. `(F_(1))/(F_(2))= (16)/(81)`
C. `(9)/(8)`
D. `(8)/(9)`

Answer

To solve the problem, we need to determine how the force between two identical short bar magnets changes when they are moved from a distance of \( d_1 = 2 \, \text{m} \) to \( d_2 = 3 \, \text{m} \). The force between two magnetic dipoles is inversely proportional to the fourth power of the distance between them.

### Step-by-Step Solution:

1. **Understand the relationship of force and distance for magnetic dipoles**:
The force \( F \) between two identical magnetic dipoles is given by the formula:
\[ F \propto \frac{1}{d^4} \]
where \( d \) is the distance between the dipoles.

2. **Define the initial and final distances**:
Let the initial distance be \( d_1 = 2 \, \text{m} \) and the final distance be \( d_2 = 3 \, \text{m} \).

3. **Set up the ratio of the forces**:
The ratio of the forces at the two distances can be expressed as:
\[ \frac{F_1}{F_2} = \frac{d_2^4}{d_1^4} \]

4. **Substitute the values of \( d_1 \) and \( d_2 \)**:
Substitute \( d_1 = 2 \, \text{m} \) and \( d_2 = 3 \, \text{m} \) into the equation:
\[ \frac{F_1}{F_2} = \frac{(3)^4}{(2)^4} \]

5. **Calculate \( (3)^4 \) and \( (2)^4 \)**:
\[ (3)^4 = 81 \quad \text{and} \quad (2)^4 = 16 \]

6. **Calculate the ratio**:
\[ \frac{F_1}{F_2} = \frac{81}{16} \]

7. **Interpret the result**:
This means that the force between the magnets changes by a factor of \( \frac{81}{16} \) when the distance is increased from 2 m to 3 m.

### Final Answer:
The force between the two identical short bar magnets changes by a factor of \( \frac{81}{16} \).