A coil in the shape of an equilateral triangle of side l is suspended between the pole pieces of a permanent magnet such that`vec B ` is in the plane of the coil. If due to a current `i` in the triangle a torque `tau` acts on it, the side `l` of the triangle is

Options

A. `(2)/(sqrt3)((tau)/(Bi))^((1)//(2))`
B. `(2)/(sqrt3)((tau)/(Bi))`
C. `(2)((tau)/(sqrt3(Bi)))^((1)//(2))`
D. `(1)/(sqrt(3))(tau)/(Bi)`

Answer

To solve the problem, we need to find the side length \( l \) of an equilateral triangle coil that experiences a torque \( \tau \) when a current \( i \) flows through it in the presence of a magnetic field \( \vec{B} \) that lies in the plane of the coil.

### Step-by-Step Solution:

1. **Understanding Torque on a Current-Carrying Coil**:
The torque \( \tau \) acting on a current-carrying coil in a magnetic field is given by the formula:
\[ \tau = I \cdot A \cdot B \cdot \sin(\theta) \]
where:
- \( I \) is the current flowing through the coil,
- \( A \) is the area of the coil,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the area vector of the coil and the magnetic field.

2. **Identifying the Angle**:
Since the magnetic field \( \vec{B} \) is in the plane of the coil, the area vector \( \vec{A} \) of the triangle will be perpendicular to the plane of the coil. Therefore, \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1 \).

3. **Substituting Values into the Torque Formula**:
The torque can now be simplified to:
\[ \tau = I \cdot A \cdot B \]

4. **Calculating the Area of the Equilateral Triangle**:
The area \( A \) of an equilateral triangle with side length \( l \) is given by:
\[ A = \frac{\sqrt{3}}{4} l^2 \]

5. **Substituting the Area into the Torque Equation**:
Substituting the area into the torque equation gives:
\[ \tau = I \cdot \left(\frac{\sqrt{3}}{4} l^2\right) \cdot B \]

6. **Rearranging to Solve for \( l^2 \)**:
Rearranging the equation to isolate \( l^2 \):
\[ l^2 = \frac{4\tau}{I \sqrt{3} B} \]

7. **Taking the Square Root to Find \( l \)**:
Taking the square root of both sides results in:
\[ l = 2 \sqrt{\frac{\tau}{I \sqrt{3} B}} \]

### Final Result:
Thus, the side length \( l \) of the equilateral triangle is given by:
\[ l = 2 \sqrt{\frac{\tau}{I \sqrt{3} B}} \]