Two coils have a mutual inductance 0.005 H. The alternating current changes in the first coil according to equation I = `underset(o)(I)`sin`omega`t, where `underset(o)(I)` = 10 A and `omega` = 100`pi` `rads^(-1)`. The maximum value of emf in the second coil
is (in volt)

Options

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

Explanation

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

### 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:**
- The induced emf in the second coil is given by the formula:
\[ \text{emf} = -M \frac{dI_1}{dt} \]
- First, we need to differentiate the current \( I(t) \):
\[ I(t) = I_0 \sin(\omega t) \]
\[ \frac{dI}{dt} = I_0 \omega \cos(\omega t) \]

3. **Substitute the Values:**
- Substitute \( I_0 \) and \( \omega \) into the differentiation result:
\[ \frac{dI}{dt} = 10 \cdot (100\pi) \cos(100\pi t) = 1000\pi \cos(100\pi t) \]

4. **Calculate the Induced emf:**
- Now substitute \( \frac{dI}{dt} \) into the emf formula:
\[ \text{emf} = -M \frac{dI}{dt} = -0.005 \cdot (1000\pi \cos(100\pi t)) \]
\[ \text{emf} = -5\pi \cos(100\pi t) \]

5. **Determine the Maximum Value of emf:**
- The maximum value of \( \cos(100\pi t) \) is 1, hence:
\[ \text{Maximum emf} = 5\pi \, \text{V} \]

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