Calculate the maximum emf induced in a coil of 100 turns and `0.01 m^(2)` area rotating at the rate of 50 rps about an axis perpendicular to a uniform magnetic field of 0.05 T. If the resistance of the coil is `30 Omega`, what is the maximum power generated by it ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Number of turns (N) = 100
- Area of the coil (A) = 0.01 m²
- Rate of rotation (f) = 50 rps (revolutions per second)
- Magnetic field (B) = 0.05 T
- Resistance of the coil (R) = 30 Ω

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 3: Calculate the maximum induced emf (ε₀)
The maximum induced emf in a coil can be calculated using the formula:
\[ \epsilon_0 = N \cdot B \cdot A \cdot \omega \]
Substituting the known values:
\[ \epsilon_0 = 100 \cdot 0.05 \cdot 0.01 \cdot (100\pi) \]
Calculating this step-by-step:
\[ \epsilon_0 = 100 \cdot 0.05 = 5 \]
\[ \epsilon_0 = 5 \cdot 0.01 = 0.05 \]
\[ \epsilon_0 = 0.05 \cdot (100\pi) \approx 0.05 \cdot 314.16 \approx 15.7 \, \text{V} \]

### Step 4: Calculate the maximum current (I₀)
The maximum current can be calculated using Ohm's law:
\[ I_0 = \frac{\epsilon_0}{R} \]
Substituting the values:
\[ I_0 = \frac{15.7}{30} \approx 0.524 \, \text{A} \]

### Step 5: Calculate the maximum power (P)
The maximum power generated can be calculated using the formula:
\[ P = \epsilon_0 \cdot I_0 \]
Substituting the values:
\[ P = 15.7 \cdot 0.524 \approx 8.23 \, \text{W} \]

### Final Answer
The maximum power generated by the coil is approximately **8.23 W**.

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