A square conducting loop of side length L carries a current I.The magnetic field at the centre of the loop is

Options

A. independent of L
B. proportional L
C. inversely proportional to L
D. linearly proportional to L

Explanation

To find the magnetic field at the center of a square conducting loop of side length \( L \) carrying a current \( I \), we can follow these steps:

### Step 1: Understand the Geometry of the Loop
- Visualize the square loop with each side of length \( L \).
- The current \( I \) flows through each side of the loop.

### Step 2: Determine the Contribution of Each Side
- The magnetic field at the center due to each side of the square can be calculated using the Biot-Savart law.
- For a straight current-carrying conductor, the magnetic field \( B \) at a distance \( r \) from the wire is given by:
\[ B = \frac{\mu_0 I}{4\pi r} \sin(\theta) \]
where \( \theta \) is the angle subtended by the segment at the point where the field is being calculated.

### Step 3: Calculate the Distance from the Center to Each Side
- The distance from the center of the square to the midpoint of any side is \( \frac{L}{2} \).

### Step 4: Calculate the Magnetic Field Contribution from One Side
- For one side of the square, the angle \( \theta \) is \( 90^\circ \) (since the field lines are perpendicular to the radius at the midpoint).
- Therefore, \( \sin(90^\circ) = 1 \).
- The magnetic field due to one side at the center is:
\[ B_1 = \frac{\mu_0 I}{4\pi \left(\frac{L}{2}\right)} = \frac{\mu_0 I}{2\pi L} \]

### Step 5: Consider All Four Sides
- Since the contributions from all four sides of the square loop are symmetrical and directed into the center, we can sum them up.
- The total magnetic field \( B \) at the center is:
\[ B = 4 \times B_1 = 4 \times \frac{\mu_0 I}{2\pi L} = \frac{2\mu_0 I}{\pi L} \]

### Step 6: Conclusion
- The magnetic field at the center of the square loop is:
\[ B = \frac{2\mu_0 I}{\pi L} \]