Two coils have a mutual inductance of `0.005 H`. the current changes in the first coil according to equation `I=I_0 sinomegat`, where `I_0=10A` and `omega=100pirad//s`. The maximum value of emf (in volt) in the second coil is

Options

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

Explanation

To solve the problem, we need to find the maximum value of the electromotive force (emf) induced in the second coil due to the changing current in the first coil. We will use the formula for mutual inductance and the relationship between current and emf.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Mutual inductance \( M = 0.005 \, H \)
- Current in the first coil \( I(t) = I_0 \sin(\omega t) \)
- Where \( I_0 = 10 \, A \) and \( \omega = 100 \pi \, \text{rad/s} \)

2. **Differentiate the Current:**
- We need to find \( \frac{dI}{dt} \).
- Given \( I(t) = I_0 \sin(\omega t) \), we differentiate it:
\[ \frac{dI}{dt} = I_0 \cdot \omega \cos(\omega t) \]

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

4. **Find the Maximum Value of \( \frac{dI}{dt} \):**
- The maximum value of \( \cos(\omega t) \) is 1.
- Therefore, the maximum value of \( \frac{dI}{dt} \) is:
\[ \left(\frac{dI}{dt}\right)_{\text{max}} = 1000\pi \]

5. **Calculate the Maximum Induced EMF:**
- The induced emf \( \mathcal{E} \) in the second coil is given by:
\[ \mathcal{E} = -M \frac{dI}{dt} \]
- Taking the maximum value:
\[ \mathcal{E}_{\text{max}} = M \left(\frac{dI}{dt}\right)_{\text{max}} = 0.005 \cdot 1000\pi \]

6. **Final Calculation:**
- Calculate \( \mathcal{E}_{\text{max}} \):
\[ \mathcal{E}_{\text{max}} = 5\pi \, \text{V} \]

### Conclusion:
The maximum value of the induced emf in the second coil is \( 5\pi \, \text{V} \).