Two coils have the mutual inductance of 0.05 H. The current changes in the first coil as `I=I_(0)sin omegat`, where `I_(0)=1A` and `omega=100pi"rad/s"`. The maximum emf induced in secondary coil is

Options

A. 2.5 V
B. 10 V
C. `6piV`
D. `5piV`

Explanation

To solve the problem of finding the maximum emf induced in the secondary coil due to the changing current in the first coil, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Mutual inductance \( M = 0.05 \, \text{H} \)
- Current in the first coil as \( I(t) = I_0 \sin(\omega t) \)
- \( I_0 = 1 \, \text{A} \)
- \( \omega = 100\pi \, \text{rad/s} \)

### Step 2: Determine the Formula for Induced EMF
The induced emf (\( \mathcal{E} \)) in the secondary coil due to the changing current in the first coil can be calculated using the formula:
\[ \mathcal{E} = -M \frac{dI}{dt} \]
where \( \frac{dI}{dt} \) is the rate of change of current in the first coil.

### Step 3: Calculate \( \frac{dI}{dt} \)
The current \( I(t) \) is given as:
\[ I(t) = I_0 \sin(\omega t) \]
To find \( \frac{dI}{dt} \), we differentiate \( I(t) \):
\[ \frac{dI}{dt} = I_0 \frac{d}{dt}(\sin(\omega t)) = I_0 \omega \cos(\omega t) \]
Substituting \( I_0 = 1 \, \text{A} \) and \( \omega = 100\pi \):
\[ \frac{dI}{dt} = 1 \cdot (100\pi) \cos(100\pi t) = 100\pi \cos(100\pi t) \]

### Step 4: Substitute \( \frac{dI}{dt} \) into the EMF Formula
Now, substituting \( \frac{dI}{dt} \) back into the emf formula:
\[ \mathcal{E} = -M \frac{dI}{dt} = -0.05 \cdot (100\pi \cos(100\pi t)) \]
This simplifies to:
\[ \mathcal{E} = -5\pi \cos(100\pi t) \]

### Step 5: Find the Maximum EMF
The maximum value of \( \cos(100\pi t) \) is 1. Therefore, the maximum emf is:
\[ \mathcal{E}_{\text{max}} = 5\pi \, \text{V} \]

### Conclusion
The maximum emf induced in the secondary coil is:
\[ \mathcal{E}_{\text{max}} = 5\pi \, \text{V} \]

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