A coil in the shape of an equilateral triangle of side `l` is suspended between the pole pieces of permanent magnet. Such that `vecB` is in plane of the coil. If due to a current I in the triangle, a torque `tau` acts on it, the side l of the triangel is:

Options

A. a) `(2)/(sqrt(3))((tau)/(Bi))`
B. b) `(1)/(sqrt(3))(tau)/(Bi)`
C. c) `2((tau)/(sqrt(3) Bi))^((1)/(2))`
D. d) `(2)/(sqrt(3))((tau)/(Bi))^((1)/(2))`

Explanation

To solve the problem, we need to determine the side length \( l \) of an equilateral triangle coil that is suspended in a magnetic field and experiences a torque \( \tau \) due to a current \( I \).

### Step-by-Step Solution:

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

2. **Setting the Angle**:
In this case, the magnetic field \( \vec{B} \) is in the plane of the coil, which means \( \theta = 90^\circ \). Therefore, \( \sin(90^\circ) = 1 \), and the torque simplifies to:
\[ \tau = I \cdot B \cdot A \]

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

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

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

6. **Taking the Square Root**:
Now, taking the square root of both sides to find \( l \):
\[ l = \sqrt{\frac{4\tau}{I \cdot B \cdot \sqrt{3}}} \]

7. **Simplifying the Expression**:
This can be simplified further:
\[ l = \frac{2\sqrt{\tau}}{\sqrt{I \cdot B \cdot \sqrt{3}}} \]

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