Calculate the maximum torque experienced by a rectangular coil of length 10 em and breadth 8 cm carrying current of 100 mA and has 1,000 turns placed in a region of magnetic field of `0.2 T` such that magnetic field lie on the plane of coil.

Answer

To calculate the maximum torque experienced by a rectangular coil in a magnetic field, we can use the formula for torque (τ) on a current-carrying coil in a magnetic field:

\[ \tau = N \cdot I \cdot A \cdot B \cdot \sin \theta \]

Where:
- \(N\) = number of turns in the coil
- \(I\) = current flowing through the coil
- \(A\) = area of the coil
- \(B\) = magnetic field strength
- \(\theta\) = angle between the magnetic field and the normal to the plane of the coil

### Step 1: Convert units
First, we need to convert all measurements to SI units:
- Length of the coil, \(L = 10 \, \text{cm} = 0.1 \, \text{m}\)
- Breadth of the coil, \(B = 8 \, \text{cm} = 0.08 \, \text{m}\)
- Current, \(I = 100 \, \text{mA} = 0.1 \, \text{A}\)
- Number of turns, \(N = 1000\)
- Magnetic field strength, \(B = 0.2 \, \text{T}\)

### Step 2: Calculate the area of the coil
The area \(A\) of the rectangular coil can be calculated using the formula:

\[ A = L \times B \]

Substituting the values:

\[ A = 0.1 \, \text{m} \times 0.08 \, \text{m} = 0.008 \, \text{m}^2 \]

### Step 3: Determine the maximum torque
The maximum torque occurs when \(\sin \theta = 1\) (i.e., when the angle \(\theta\) is 90 degrees). Thus, the formula for maximum torque simplifies to:

\[ \tau_{\text{max}} = N \cdot I \cdot A \cdot B \]

Substituting the values we have:

\[ \tau_{\text{max}} = 1000 \cdot 0.1 \cdot 0.008 \cdot 0.2 \]

Calculating this step-by-step:

1. Calculate \(N \cdot I\):
\[ 1000 \cdot 0.1 = 100 \]

2. Calculate \(A \cdot B\):
\[ 0.008 \cdot 0.2 = 0.0016 \]

3. Now multiply these results:
\[ \tau_{\text{max}} = 100 \cdot 0.0016 = 0.16 \, \text{N m} \]

### Final Answer
The maximum torque experienced by the rectangular coil is:

\[ \tau_{\text{max}} = 0.16 \, \text{N m} \]

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