A rectangular coil of 10 em by 8 cm is suspended vertically in a region of horizontal magnetic field of `1.5 `T such that the field lines make an angle of `60^(@)` with the normal to the coil. Calculate the couple acting on the coil when a current of 5 A is passing through it.

Answer

To calculate the couple (torque) acting on the rectangular coil when a current is passing through it, we can follow these steps:

### Step 1: Understand the parameters given in the problem.
- Dimensions of the rectangular coil:
- Length (l) = 10 cm = 0.1 m
- Breadth (b) = 8 cm = 0.08 m
- Magnetic field (B) = 1.5 T
- Current (I) = 5 A
- Angle (θ) between the magnetic field and the normal to the coil = 60°

### Step 2: Calculate the area (A) of the coil.
The area 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: Calculate the torque (τ) acting on the coil.
The torque (τ) acting on the coil in a magnetic field can be calculated using the formula:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
Where:
- n = number of turns (since it is not mentioned, we assume n = 1)
- I = current (5 A)
- A = area (0.008 m²)
- B = magnetic field (1.5 T)
- θ = angle (60°)

Substituting the values:
\[ \tau = 1 \cdot 5 \, \text{A} \cdot 0.008 \, \text{m}^2 \cdot 1.5 \, \text{T} \cdot \sin(60^\circ) \]

### Step 4: Calculate \(\sin(60^\circ)\).
The value of \(\sin(60^\circ)\) is:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute \(\sin(60^\circ)\) into the torque equation.
Now substituting \(\sin(60^\circ)\):
\[ \tau = 1 \cdot 5 \cdot 0.008 \cdot 1.5 \cdot \frac{\sqrt{3}}{2} \]

### Step 6: Calculate the torque.
Calculating the values:
\[ \tau = 5 \cdot 0.008 \cdot 1.5 \cdot \frac{\sqrt{3}}{2} \]
\[ \tau = 0.06 \cdot \frac{\sqrt{3}}{2} \]
\[ \tau = 0.03\sqrt{3} \]

Using \(\sqrt{3} \approx 1.732\):
\[ \tau \approx 0.03 \cdot 1.732 \approx 0.052 \, \text{Nm} \]

### Final Answer:
The couple acting on the coil is approximately \(0.052 \, \text{Nm}\).

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