Two coils have a mutual inductance `0.005 H`. The current changes in the first coil according to equation `I=I_(0)sin omegat`, where `I_(0)=10A` and `omega=100pi` radian/sec. The maximum value of e.m.f. in the second coil is

Options

A. `2pi`
B. `5pi`
C. `pi`
D. `4pi`

Answer

To find the maximum value of the induced electromotive force (e.m.f.) in the second coil due to the changing current in the first coil, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Mutual inductance, \( M = 0.005 \, \text{H} \)
- Current in the first coil, \( I(t) = I_0 \sin(\omega t) \)
- Maximum current, \( I_0 = 10 \, \text{A} \)
- Angular frequency, \( \omega = 100\pi \, \text{rad/s} \)

2. **Differentiate the Current**:
- To find the induced e.m.f., we need to calculate the rate of change of current, \( \frac{dI}{dt} \).
- Start with the expression for current:
\[ I(t) = I_0 \sin(\omega t) \]
- Differentiate with respect to time \( t \):
\[ \frac{dI}{dt} = I_0 \omega \cos(\omega t) \]

3. **Substitute the Values**:
- Substitute \( I_0 = 10 \, \text{A} \) and \( \omega = 100\pi \, \text{rad/s} \):
\[ \frac{dI}{dt} = 10 \cdot (100\pi) \cos(100\pi t) = 1000\pi \cos(100\pi t) \]

4. **Calculate the Maximum Value of the Rate of Change of Current**:
- The maximum value of \( \cos(100\pi t) \) is \( 1 \):
\[ \left(\frac{dI}{dt}\right)_{\text{max}} = 1000\pi \]

5. **Use the Formula for Induced e.m.f.**:
- The induced e.m.f. in the second coil is given by:
\[ \text{e.m.f.} = -M \frac{dI}{dt} \]
- Therefore, the maximum e.m.f. is:
\[ \text{e.m.f.}_{\text{max}} = M \left(\frac{dI}{dt}\right)_{\text{max}} = 0.005 \cdot 1000\pi \]

6. **Calculate the Final Value**:
- Now calculate the maximum e.m.f.:
\[ \text{e.m.f.}_{\text{max}} = 5\pi \, \text{V} \]

### Final Answer:
The maximum value of e.m.f. in the second coil is \( 5\pi \, \text{V} \).

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