A wire of length l is bent in the form a circular coil of some turns. A current I flows through the coil. The coil is placed in a uniform magnetic field B. The maximum torqur on the coil can be

Options

A. `(iBl^(2))/(4pi)`
B. `(iBl^(2))/(pi)`
C. `(iBl^(2))/(2pi)`
D. `(2iBl^(2))/(pi)`

Answer

To solve the problem step by step, we will derive the maximum torque on a circular coil of wire placed in a uniform magnetic field.

### Step 1: Understand the Problem
We have a wire of length \( L \) that is bent into a circular coil with \( n \) turns. A current \( I \) flows through the coil, which is placed in a uniform magnetic field \( B \). We need to find the maximum torque \( \tau \) acting on this coil.

### Step 2: Relate Length of Wire to the Coil
The total length of the wire is equal to the circumference of the coil multiplied by the number of turns:
\[ L = n \cdot (2\pi r) \]
where \( r \) is the radius of the coil.

### Step 3: Solve for Radius
From the equation above, we can express the radius \( r \) in terms of \( L \) and \( n \):
\[ r = \frac{L}{2\pi n} \]

### Step 4: Calculate the Area of the Coil
The area \( A \) of the circular coil can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting \( r \) from the previous step:
\[ A = \pi \left(\frac{L}{2\pi n}\right)^2 = \frac{L^2}{4\pi n^2} \]

### Step 5: Calculate the Magnetic Moment
The magnetic moment \( m \) of the coil is given by:
\[ m = n \cdot I \cdot A \]
Substituting the expression for area \( A \):
\[ m = n \cdot I \cdot \frac{L^2}{4\pi n^2} = \frac{I L^2}{4\pi n} \]

### Step 6: Calculate the Torque
The torque \( \tau \) on the coil in a magnetic field is given by:
\[ \tau = m \cdot B \cdot \sin(\theta) \]
where \( \theta \) is the angle between the magnetic moment and the magnetic field. For maximum torque, \( \sin(\theta) = 1 \) (i.e., \( \theta = 90^\circ \)):
\[ \tau_{\text{max}} = m \cdot B \]
Substituting the expression for \( m \):
\[ \tau_{\text{max}} = \left(\frac{I L^2}{4\pi n}\right) \cdot B \]

### Step 7: Maximize the Torque
To maximize the torque, we need to minimize \( n \). The minimum value of \( n \) is 1 (i.e., a single turn):
\[ \tau_{\text{max}} = \frac{I L^2}{4\pi} \cdot B \]

### Final Result
Thus, the maximum torque on the coil is:
\[ \tau_{\text{max}} = \frac{I L^2 B}{4\pi} \]