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. 10 mH
B. 15 mH
C. 25 mH
D. 5 mH

Explanation

To find the mutual inductance between the two coils, we can follow these steps:

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

### Step 2: Relate the induced e.m.f. to mutual inductance
The induced e.m.f. (\(E\)) in the second coil due to the changing current in the first coil can be expressed using the formula:
\[ E = -M \frac{dI}{dt} \]
where \(M\) is the mutual inductance and \(\frac{dI}{dt}\) is the rate of change of current in the first coil.

### Step 3: Differentiate the current function
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, so:
\[ \left(\frac{dI}{dt}\right)_{\text{max}} = 1000 \pi \]

### Step 5: Substitute into the induced e.m.f. equation
Now we can substitute the maximum value of \(\frac{dI}{dt}\) into the e.m.f. equation:
\[ E_{\text{max}} = M \cdot \left(\frac{dI}{dt}\right)_{\text{max}} \]
Substituting the known values:
\[ 5 \pi = M \cdot 1000 \pi \]

### Step 6: Solve for mutual inductance \(M\)
Dividing both sides by \(1000 \pi\):
\[ M = \frac{5 \pi}{1000 \pi} = \frac{5}{1000} = \frac{1}{200} \, \text{H} \]

### Step 7: Convert to milliHenries
To express the mutual inductance in milliHenries:
\[ M = \frac{1}{200} \, \text{H} = 0.005 \, \text{H} = 5 \, \text{mH} \]

### Final Answer
The mutual inductance between the coils is:
\[ M = 5 \, \text{mH} \]

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