A square frame of side `l` carries a current produces a field `B` at its centre. The same current is passed through a circular loop having same perimeter as the square. The field at its centre is `B'`, the ratio of `B//B'` is

Options

A. `(8)/(pi^(2))`
B. `(8sqrt(2))/(pi^(2))`
C. `(16)/(pi^(2))`
D. `(16)/(sqrt(2)pi^(2))`

Answer

To solve the problem, we need to find the ratio of the magnetic field \( B \) produced by a square frame carrying a current \( I \) at its center to the magnetic field \( B' \) produced by a circular loop with the same perimeter carrying the same current \( I \).

### Step-by-Step Solution:

1. **Determine the Perimeter of the Square Frame:**
The perimeter \( P \) of a square frame with side length \( l \) is given by:
\[ P = 4l \]

2. **Calculate the Magnetic Field \( B \) at the Center of the Square Frame:**
The magnetic field \( B \) at the center of a square carrying current \( I \) can be calculated using the formula:
\[ B = \frac{4 \mu_0 I}{\pi l} \]
where \( \mu_0 \) is the permeability of free space.

3. **Determine the Radius of the Circular Loop:**
Since the circular loop has the same perimeter as the square, we can set the perimeter of the circular loop equal to that of the square:
\[ 2\pi r = 4l \]
Solving for \( r \):
\[ r = \frac{4l}{2\pi} = \frac{2l}{\pi} \]

4. **Calculate the Magnetic Field \( B' \) at the Center of the Circular Loop:**
The magnetic field \( B' \) at the center of a circular loop carrying current \( I \) is given by:
\[ B' = \frac{\mu_0 I}{2r} \]
Substituting \( r \) from the previous step:
\[ B' = \frac{\mu_0 I}{2 \cdot \frac{2l}{\pi}} = \frac{\mu_0 I \pi}{4l} \]

5. **Find the Ratio \( \frac{B}{B'} \):**
Now we can find the ratio of the magnetic fields:
\[ \frac{B}{B'} = \frac{\frac{4 \mu_0 I}{\pi l}}{\frac{\mu_0 I \pi}{4l}} \]
Simplifying this expression:
\[ \frac{B}{B'} = \frac{4 \mu_0 I}{\pi l} \cdot \frac{4l}{\mu_0 I \pi} = \frac{16}{\pi^2} \]

6. **Final Result:**
Thus, the ratio of the magnetic field at the center of the square frame to that at the center of the circular loop is:
\[ \frac{B}{B'} = \frac{16}{\pi^2} \]