A coil of 100 turns, `5cm^(2)` area is placed in external magnetic field of 0.2 Tesla (S.I) in such a way that plane of the coil makes an angle `30^(@)` with the field direction. Calculate magnetic flux of the coil (in weber)

Options

A. `5xx10^(-3)`
B. `4xx10^(-3)`
C. `1.2xx10^(-3)`
D. `3xx10^(-3)`

Answer

To calculate the magnetic flux through a coil placed in a magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Number of turns in the coil (N) = 100 turns
- Area of the coil (A) = 5 cm² = \(5 \times 10^{-4}\) m² (conversion from cm² to m²)
- Magnetic field strength (B) = 0.2 T (Tesla)
- Angle (θ) between the magnetic field and the normal to the coil = 30°

### Step 2: Calculate the angle between the magnetic field and the area vector
The area vector is always perpendicular to the surface of the coil. Therefore, if the angle between the plane of the coil and the magnetic field is 30°, the angle between the magnetic field and the area vector (θ') is:
\[ θ' = 90° - θ = 90° - 30° = 60° \]

### Step 3: Use the formula for magnetic flux
The magnetic flux (Φ) through the coil can be calculated using the formula:
\[ Φ = N \cdot B \cdot A \cdot \cos(θ') \]
Where:
- Φ = magnetic flux in Weber
- N = number of turns
- B = magnetic field strength
- A = area of the coil
- θ' = angle between the magnetic field and the area vector

### Step 4: Substitute the values into the formula
Now we can substitute the known values into the formula:
\[ Φ = 100 \cdot 0.2 \cdot (5 \times 10^{-4}) \cdot \cos(60°) \]

### Step 5: Calculate cos(60°)
The cosine of 60 degrees is:
\[ \cos(60°) = \frac{1}{2} \]

### Step 6: Substitute cos(60°) into the equation
Now we can substitute this value back into the equation:
\[ Φ = 100 \cdot 0.2 \cdot (5 \times 10^{-4}) \cdot \frac{1}{2} \]

### Step 7: Perform the calculations
Calculating step by step:
1. Calculate \(0.2 \cdot \frac{1}{2} = 0.1\)
2. Now, substitute back:
\[ Φ = 100 \cdot 0.1 \cdot (5 \times 10^{-4}) \]
3. Calculate \(100 \cdot 0.1 = 10\)
4. Finally, calculate:
\[ Φ = 10 \cdot (5 \times 10^{-4}) = 5 \times 10^{-3} \text{ Weber} \]

### Final Answer
The magnetic flux through the coil is:
\[ Φ = 5 \times 10^{-3} \text{ Weber} \]

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