A circular coil of 30 turns and radius 15 cm carries a current of 5 A. Calculate the net torque on the coil when it is placed in a uniform magnetic field of` 0.2` T normal to the plane of coil. Also calculate the net force on the coil.

Answer

To solve the problem, we will calculate the net torque on a circular coil placed in a uniform magnetic field and the net force acting on it.

### Given Data:
- Number of turns (N) = 30
- Radius of the coil (r) = 15 cm = 0.15 m
- Current (I) = 5 A
- Magnetic field (B) = 0.2 T

### Step 1: Calculate the Area of the Coil
The area (A) of a circular coil can be calculated using the formula:
\[ A = \pi r^2 \]

Substituting the value of radius:
\[ A = \pi (0.15)^2 \]
\[ A = \pi (0.0225) \]
\[ A \approx 0.0707 \, \text{m}^2 \]

### Step 2: Calculate the Torque on the Coil
The torque (\(\tau\)) on a coil in a magnetic field is given by the formula:
\[ \tau = N \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
Where:
- \(\theta\) is the angle between the magnetic field and the normal to the coil's plane.

Since the magnetic field is normal to the plane of the coil, \(\theta = 0^\circ\) and \(\sin(0) = 0\).

Thus,
\[ \tau = N \cdot I \cdot A \cdot B \cdot \sin(0) \]
\[ \tau = N \cdot I \cdot A \cdot B \cdot 0 \]
\[ \tau = 0 \]

### Step 3: Calculate the Net Force on the Coil
The net force (F) on the coil can be calculated using the formula:
\[ F = N \cdot I \cdot L \cdot B \cdot \sin(\alpha) \]
Where:
- \(L\) is the length of the wire in the coil.
- \(\alpha\) is the angle between the wire and the magnetic field.

The length of the wire for one turn of the coil is the circumference:
\[ L = 2\pi r = 2\pi (0.15) \]
\[ L \approx 0.9425 \, \text{m} \]

Since the magnetic field is perpendicular to the current in the wire, \(\alpha = 90^\circ\) and \(\sin(90) = 1\).

Thus,
\[ F = N \cdot I \cdot L \cdot B \cdot \sin(90) \]
\[ F = 30 \cdot 5 \cdot 0.9425 \cdot 0.2 \cdot 1 \]
\[ F = 30 \cdot 5 \cdot 0.9425 \cdot 0.2 \]
\[ F = 30 \cdot 5 \cdot 0.1885 \]
\[ F = 30 \cdot 0.9425 \]
\[ F \approx 28.275 \, \text{N} \]

### Final Results:
- Net Torque on the coil: \( \tau = 0 \, \text{N m} \)
- Net Force on the coil: \( F \approx 28.275 \, \text{N} \)