Current `i` is carried in a wire of length `L`. If the wire is turned into a circular coil, the maximum magnitude of torque in a given magnetic field `B` will be

Options

A. `(LiB^(2))/(2)`
B. `(Li^(2)B)/(2)`
C. `(L^(2)iB)/(4pi)`
D. `(Li^(2)B)/(4pi)`

Explanation

To find the maximum magnitude of torque when a wire of length \( L \) carrying current \( i \) is turned into a circular coil in a magnetic field \( B \), we can follow these steps:

### Step 1: Determine the radius of the circular coil
When the wire of length \( L \) is bent into a circular shape, the circumference of the circle will equal the length of the wire. Therefore, we can express the radius \( R \) of the coil as:
\[ C = 2\pi R = L \implies R = \frac{L}{2\pi} \]

### Step 2: Calculate the magnetic moment \( \mu \)
The magnetic moment \( \mu \) of a circular coil is given by the formula:
\[ \mu = i \cdot A \]
where \( A \) is the area of the circular coil. The area \( A \) can be calculated using the radius \( R \):
\[ A = \pi R^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \frac{L^2}{4\pi} \]
Thus, substituting this back into the magnetic moment formula gives:
\[ \mu = i \cdot \frac{L^2}{4\pi} \]

### Step 3: Calculate the torque \( \tau \)
The torque \( \tau \) experienced by the coil in a magnetic field \( B \) is given by the formula:
\[ \tau = \mu B \sin \theta \]
For maximum torque, the angle \( \theta \) between the magnetic moment \( \mu \) and the magnetic field \( B \) should be \( 90^\circ \), thus \( \sin 90^\circ = 1 \):
\[ \tau_{\text{max}} = \mu B = \left(i \cdot \frac{L^2}{4\pi}\right) B \]

### Final Result
Therefore, the maximum magnitude of torque is:
\[ \tau_{\text{max}} = \frac{i L^2 B}{4\pi} \]