A straight wire is first bent into a circle of radius 'r' and then into a square of side 'x' each of oneturn. If currents flowing through them are in the ratio 4 : 5, the ratio of their effective magnetic moments is

Options

A. `pi/8`
B. `12/(5pi)`
C. `16/(5pi)`
D. `8/(5pi)`

Answer

To solve the problem, we need to find the ratio of the effective magnetic moments of a wire bent into a circle and a square, given the currents flowing through them in the ratio of 4:5.

### Step-by-Step Solution:

1. **Determine the Length of the Wire:**
Let the length of the wire be \( l \).

2. **Wire Bent into a Circle:**
- The circumference of the circle is given by:
\[ l = 2\pi r \]
- From this, we can express the radius \( r \) in terms of \( l \):
\[ r = \frac{l}{2\pi} \]
- The area \( A_1 \) of the circle is:
\[ A_1 = \pi r^2 = \pi \left(\frac{l}{2\pi}\right)^2 = \frac{l^2}{4\pi} \]

3. **Wire Bent into a Square:**
- The perimeter of the square is given by:
\[ 4x = l \implies x = \frac{l}{4} \]
- The area \( A_2 \) of the square is:
\[ A_2 = x^2 = \left(\frac{l}{4}\right)^2 = \frac{l^2}{16} \]

4. **Magnetic Moment Calculation:**
- The magnetic moment \( M \) for a loop is given by:
\[ M = I \cdot A \]
- For the circle, the magnetic moment \( M_1 \) is:
\[ M_1 = I_1 \cdot A_1 = I_1 \cdot \frac{l^2}{4\pi} \]
- For the square, the magnetic moment \( M_2 \) is:
\[ M_2 = I_2 \cdot A_2 = I_2 \cdot \frac{l^2}{16} \]

5. **Current Ratio:**
- Given that the currents are in the ratio:
\[ \frac{I_1}{I_2} = \frac{4}{5} \]
- We can express \( I_1 \) in terms of \( I_2 \):
\[ I_1 = \frac{4}{5} I_2 \]

6. **Ratio of Magnetic Moments:**
- Now substituting \( I_1 \) into the expression for \( M_1 \):
\[ M_1 = \left(\frac{4}{5} I_2\right) \cdot \frac{l^2}{4\pi} = \frac{4 I_2 l^2}{20\pi} = \frac{I_2 l^2}{5\pi} \]
- Now we can find the ratio of the magnetic moments:
\[ \frac{M_1}{M_2} = \frac{\frac{I_2 l^2}{5\pi}}{I_2 \cdot \frac{l^2}{16}} = \frac{1}{5\pi} \cdot \frac{16}{l^2} = \frac{16}{5\pi} \]

7. **Final Result:**
- Therefore, the ratio of the effective magnetic moments is:
\[ \frac{M_1}{M_2} = \frac{16}{5\pi} \]

### Conclusion:
The ratio of their effective magnetic moments is \( \frac{16}{5\pi} \).