A current `I = 10 sin (100pi t)` amp. Is passed in first coil, which induces a maximum e.m.f of `5pi` volt in second coil. The mutual inductance between the coils is-

Options

A. 5 mH
B. 10 mH
C. 15 mH
D. 20 mH

Answer

To find the mutual inductance between the two coils given the current in the first coil and the induced e.m.f. in the second coil, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given parameters**:
- The current in the first coil is given by \( I(t) = 10 \sin(100\pi t) \) A.
- The maximum induced e.m.f. in the second coil is \( E_{\text{max}} = 5\pi \) V.

2. **Extract the amplitude and angular frequency from the current equation**:
- From the equation \( I(t) = I_0 \sin(\omega t) \), we can identify:
- \( I_0 = 10 \) A (amplitude of the current)
- \( \omega = 100\pi \) rad/s (angular frequency)

3. **Use the formula for induced e.m.f.**:
- The induced e.m.f. in the second coil due to the changing current in the first coil is given by:
\[ E = M \frac{dI}{dt} \]
- Here, \( M \) is the mutual inductance and \( \frac{dI}{dt} \) is the rate of change of current.

4. **Differentiate the current with respect to time**:
- To find \( \frac{dI}{dt} \):
\[ \frac{dI}{dt} = \frac{d}{dt}(10 \sin(100\pi t)) = 10 \cdot 100\pi \cos(100\pi t) = 1000\pi \cos(100\pi t) \]

5. **Find the maximum value of \( \frac{dI}{dt} \)**:
- The maximum value occurs when \( \cos(100\pi t) = 1 \):
\[ \left(\frac{dI}{dt}\right)_{\text{max}} = 1000\pi \text{ A/s} \]

6. **Relate the maximum e.m.f. to mutual inductance**:
- At maximum condition:
\[ E_{\text{max}} = M \cdot \left(\frac{dI}{dt}\right)_{\text{max}} \]
- Substituting the known values:
\[ 5\pi = M \cdot 1000\pi \]

7. **Solve for mutual inductance \( M \)**:
- Dividing both sides by \( 1000\pi \):
\[ M = \frac{5\pi}{1000\pi} = \frac{5}{1000} = 0.005 \text{ H} = 5 \text{ mH} \]

### Final Answer:
The mutual inductance between the coils is \( M = 5 \text{ mH} \).