A coil bent in form of an equilateral triangle of side 10 cm is suspended with the help of a mass less string through one of its vertex. Calculate the torque experienced by the coil if the magnetic field in the region is` 0.2` T directed horizontally and current of 1 A is passing through the coil.

Answer

To calculate the torque experienced by a coil bent in the form of an equilateral triangle in a magnetic field, we can follow these steps:

### Step 1: Understand the given parameters
- Side of the equilateral triangle, \( a = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Current through the coil, \( I = 1 \, \text{A} \)
- Magnetic field strength, \( B = 0.2 \, \text{T} \)

### Step 2: Calculate the area of the equilateral triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting the value of \( a \):
\[ A = \frac{\sqrt{3}}{4} (0.1)^2 = \frac{\sqrt{3}}{4} \times 0.01 = \frac{\sqrt{3}}{400} \, \text{m}^2 \]

### Step 3: Calculate the torque
The torque \( \tau \) experienced by the coil in a magnetic field is given by the formula:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
where:
- \( n \) is the number of turns (since it's a single coil, \( n = 1 \))
- \( \theta \) is the angle between the area vector and the magnetic field vector. For maximum torque, we take \( \theta = 90^\circ \) (thus \( \sin(90^\circ) = 1 \)).

Substituting the values:
\[ \tau = 1 \cdot 1 \cdot \left(\frac{\sqrt{3}}{400}\right) \cdot 0.2 \cdot 1 \]
Calculating this:
\[ \tau = \frac{\sqrt{3}}{400} \cdot 0.2 = \frac{0.2\sqrt{3}}{400} = \frac{\sqrt{3}}{2000} \, \text{N m} \]

### Step 4: Calculate the numerical value of torque
Using the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ \tau \approx \frac{1.732}{2000} \approx 0.000866 \, \text{N m} = 8.66 \times 10^{-4} \, \text{N m} \]

### Step 5: Direction of the torque
The direction of the torque can be determined using the right-hand rule. If the magnetic field is directed horizontally and the area vector is perpendicular to the plane of the coil, the torque will act in a direction perpendicular to both the area vector and the magnetic field.

### Final Answer
The torque experienced by the coil is approximately:
\[ \tau \approx 8.66 \times 10^{-4} \, \text{N m} \]

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