A wire is bent to form a circular coil of radius 5 cm with 20 loops. The coil is placed in a magnetic field of `0.5` T and a current of 1 A flow through it. Calculate the maximum torque exerted by magnetic field.

Answer

To calculate the maximum torque exerted by a magnetic field on a circular coil, we can follow these steps:

### Step 1: Identify the given values
- Radius of the coil, \( r = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Number of loops, \( n = 20 \)
- Current flowing through the coil, \( I = 1 \, \text{A} \)
- Magnetic field strength, \( B = 0.5 \, \text{T} \)

### Step 2: Calculate the area of one loop of the coil
The area \( A \) of a circular loop is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.05)^2 = \pi (0.0025) \approx 0.007854 \, \text{m}^2 \]

### Step 3: Calculate the magnetic moment \( m \) of the coil
The magnetic moment \( m \) is given by:
\[ m = n \cdot I \cdot A \]
Substituting the values:
\[ m = 20 \cdot 1 \cdot 0.007854 \approx 0.15708 \, \text{A m}^2 \]

### Step 4: Calculate the maximum torque \( \tau_{\text{max}} \)
The maximum torque \( \tau_{\text{max}} \) is given by the formula:
\[ \tau_{\text{max}} = m \cdot B \]
Substituting the values:
\[ \tau_{\text{max}} = 0.15708 \cdot 0.5 \approx 0.07854 \, \text{N m} \]

### Step 5: Round the answer
Rounding the result, we find:
\[ \tau_{\text{max}} \approx 0.0785 \, \text{N m} \]

### Final Answer
The maximum torque exerted by the magnetic field on the coil is approximately \( 0.0785 \, \text{N m} \).

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