A 50 turns circular coil has a radius of 3cm , it is kept in a magnetic field acting normal to the area of the coil. The magnetic field B increased from 0.10 tesla to 0.35 tesla in 2 milliseconds . The average induced e.m.f. in the coil is

Options

A. `1.77 "volts"`
B. `17.7 "volts"`
C. `177 "volts"`
D. `0.177 "volts"`

Answer

To solve the problem of finding the average induced e.m.f. in a circular coil when the magnetic field changes, we can follow these steps:

### Step 1: Identify the given values
- Number of turns (N) = 50
- Radius of the coil (r) = 3 cm = 0.03 m (convert to meters)
- Initial magnetic field (B1) = 0.10 T
- Final magnetic field (B2) = 0.35 T
- Time interval (t) = 2 ms = 2 × 10^(-3) s (convert to seconds)

### Step 2: 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 radius:
\[ A = \pi (0.03)^2 \]
\[ A = \pi (0.0009) \]
\[ A \approx 0.002827 \, \text{m}^2 \]

### Step 3: Calculate the change in magnetic flux
The magnetic flux (Φ) through the coil is given by:
\[ \Phi = N \cdot B \cdot A \]
- For initial flux (Φ1):
\[ \Phi_1 = N \cdot B_1 \cdot A = 50 \cdot 0.10 \cdot 0.002827 \]
\[ \Phi_1 \approx 0.14135 \, \text{Wb} \]

- For final flux (Φ2):
\[ \Phi_2 = N \cdot B_2 \cdot A = 50 \cdot 0.35 \cdot 0.002827 \]
\[ \Phi_2 \approx 0.24745 \, \text{Wb} \]

### Step 4: Calculate the change in magnetic flux
The change in magnetic flux (ΔΦ) is:
\[ \Delta \Phi = \Phi_2 - \Phi_1 \]
\[ \Delta \Phi \approx 0.24745 - 0.14135 \]
\[ \Delta \Phi \approx 0.1061 \, \text{Wb} \]

### Step 5: Calculate the average induced e.m.f.
The average induced e.m.f. (E) can be calculated using Faraday's law:
\[ E = \frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ E = \frac{0.1061}{2 \times 10^{-3}} \]
\[ E \approx 53.05 \, \text{V} \]

### Step 6: Finalize the answer
The average induced e.m.f. in the coil is approximately 53.05 V.