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`

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 use the formula for mutual inductance. The induced e.m.f. in the second coil can be expressed as:

\[ \text{e.m.f.} = -M \frac{dI}{dt} \]

where:
- \( M \) is the mutual inductance,
- \( I \) is the current in the first coil.

### Step 1: Identify the given values
- Mutual inductance \( M = 0.005 \, \text{H} \)
- Maximum current \( I_0 = 10 \, \text{A} \)
- Angular frequency \( \omega = 100\pi \, \text{rad/s} \)

### Step 2: Write the expression for current
The current in the first coil varies with time as:
\[ I(t) = I_0 \sin(\omega t) = 10 \sin(100\pi t) \]

### Step 3: Differentiate the current with respect to time
To find \( \frac{dI}{dt} \), we differentiate \( I(t) \):
\[ \frac{dI}{dt} = \frac{d}{dt}(10 \sin(100\pi t)) = 10 \cdot 100\pi \cos(100\pi t) = 1000\pi \cos(100\pi t) \]

### Step 4: Find the maximum value of \( \frac{dI}{dt} \)
The maximum value of \( \cos(100\pi t) \) is 1. Therefore, the maximum value of \( \frac{dI}{dt} \) is:
\[ \left( \frac{dI}{dt} \right)_{\text{max}} = 1000\pi \]

### Step 5: Calculate the maximum e.m.f. in the second coil
Now, substituting \( M \) and \( \left( \frac{dI}{dt} \right)_{\text{max}} \) into the e.m.f. formula:
\[ \text{e.m.f.}_{\text{max}} = -M \left( \frac{dI}{dt} \right)_{\text{max}} = -0.005 \times 1000\pi \]
\[ \text{e.m.f.}_{\text{max}} = -5\pi \, \text{V} \]

Since we are interested in the magnitude of the e.m.f., we take the absolute value:
\[ \text{e.m.f.}_{\text{max}} = 5\pi \, \text{V} \]

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

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