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

Explanation

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

### Step 1: Understand the relationship between mutual inductance and e.m.f.
The e.m.f. (E) induced in the second coil due to the changing current in the first coil is given by the formula:
\[ E = -M \frac{dI_1}{dt} \]
where:
- \( E \) is the induced e.m.f. in the second coil,
- \( M \) is the mutual inductance,
- \( I_1 \) is the current in the first coil.

### Step 2: Identify the current in the first coil
The current in the first coil varies with time according to the equation:
\[ I_1 = I_0 \sin(\omega t) \]
where:
- \( I_0 = 10 \, \text{A} \) (maximum current),
- \( \omega = 100\pi \, \text{rad/s} \).

### Step 3: Differentiate the current with respect to time
To find \( \frac{dI_1}{dt} \), we differentiate \( I_1 \):
\[ \frac{dI_1}{dt} = \frac{d}{dt}(I_0 \sin(\omega t)) = I_0 \omega \cos(\omega t) \]
Substituting the values of \( I_0 \) and \( \omega \):
\[ \frac{dI_1}{dt} = 10 \cdot (100\pi) \cos(100\pi t) = 1000\pi \cos(100\pi t) \]

### Step 4: Substitute into the e.m.f. formula
Now substituting \( \frac{dI_1}{dt} \) back into the e.m.f. formula:
\[ E = -M \cdot \frac{dI_1}{dt} = -0.005 \cdot (1000\pi \cos(100\pi t)) \]
This simplifies to:
\[ E = -5\pi \cos(100\pi t) \]

### Step 5: Find the maximum value of e.m.f.
The maximum value of \( E \) occurs when \( \cos(100\pi t) = 1 \):
\[ E_{\text{max}} = 5\pi \]

### Conclusion
The maximum value of the e.m.f. in the second coil is:
\[ \boxed{5\pi \, \text{V}} \]

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