A current `I=20sin(100pit)A` is passed in the first coil, which induces a maximum emf of `10piV` in the second coil. The mutual inductance for the pair of coils is

Options

A. (a)10 mH
B. (b)15 mH
C. (c)25 mH
D. (d)5 mH

Answer

To find the mutual inductance \( M \) between the two coils, we start with the information given in the problem:

1. The current in the first coil is given by:
\[ I = 20 \sin(100 \pi t) \, \text{A} \]
Here, we can identify:
- \( I_0 = 20 \) A (the maximum current)
- \( \omega = 100 \pi \) rad/s (the angular frequency)

2. The maximum induced EMF in the second coil is given as:
\[ E_{\text{max}} = 10 \pi \, \text{V} \]

### Step 1: Use the formula for induced EMF
The induced EMF \( E \) in the second coil due to the changing current in the first coil can be expressed as:
\[ E_{\text{max}} = M \frac{dI}{dt} \]
where \( M \) is the mutual inductance.

### Step 2: Differentiate the current
To find \( \frac{dI}{dt} \), we differentiate the expression for current \( I \):
\[ I = 20 \sin(100 \pi t) \]
Differentiating with respect to time \( t \):
\[ \frac{dI}{dt} = 20 \cdot 100 \pi \cos(100 \pi t) = 2000 \pi \cos(100 \pi t) \]
The maximum value of \( \frac{dI}{dt} \) occurs when \( \cos(100 \pi t) = 1 \):
\[ \left(\frac{dI}{dt}\right)_{\text{max}} = 2000 \pi \, \text{A/s} \]

### Step 3: Substitute into the EMF formula
Now we can substitute this maximum rate of change of current into the EMF formula:
\[ E_{\text{max}} = M \cdot (2000 \pi) \]
Given that \( E_{\text{max}} = 10 \pi \):
\[ 10 \pi = M \cdot (2000 \pi) \]

### Step 4: Solve for mutual inductance \( M \)
Dividing both sides by \( \pi \):
\[ 10 = M \cdot 2000 \]
Now, solving for \( M \):
\[ M = \frac{10}{2000} = \frac{1}{200} = 0.005 \, \text{H} = 5 \, \text{mH} \]

### Conclusion
Thus, the mutual inductance for the pair of coils is:
\[ \boxed{5 \, \text{mH}} \]