Two coils have a mutual inductance 0.005H. The current changes in the first coil according to the equation `I=I_(0) sin omegat "where" I_(0)=10A and omega =100pi rad//s` . The maximum value of emf wiin second coil is (pi//x)` volts. Find the value of x.

Answer

To solve the problem step by step, we will follow the principles of mutual inductance and the relationship between current and induced electromotive force (emf).

### Step 1: Understand the given information
We have:
- Mutual inductance, \( M = 0.005 \, \text{H} \)
- Current in the first coil, \( I(t) = I_0 \sin(\omega t) \)
- Maximum current, \( I_0 = 10 \, \text{A} \)
- Angular frequency, \( \omega = 100\pi \, \text{rad/s} \)

### Step 2: Find the expression for \( \frac{di}{dt} \)
The current changes with time as:
\[ I(t) = I_0 \sin(\omega t) \]

To find \( \frac{di}{dt} \), we differentiate \( I(t) \) with respect to time \( t \):
\[ \frac{di}{dt} = \frac{d}{dt}(I_0 \sin(\omega t)) = I_0 \omega \cos(\omega t) \]

### Step 3: Determine the maximum value of \( \frac{di}{dt} \)
The maximum value of \( \cos(\omega t) \) is 1, so:
\[ \left( \frac{di}{dt} \right)_{\text{max}} = I_0 \omega \]

Substituting the values of \( I_0 \) and \( \omega \):
\[ \left( \frac{di}{dt} \right)_{\text{max}} = 10 \, \text{A} \times 100\pi \, \text{rad/s} = 1000\pi \, \text{A/s} \]

### Step 4: Calculate the maximum emf in the second coil
The maximum emf \( E_{\text{max}} \) induced in the second coil due to mutual inductance is given by:
\[ E_{\text{max}} = M \left( \frac{di}{dt} \right)_{\text{max}} \]

Substituting \( M \) and \( \left( \frac{di}{dt} \right)_{\text{max}} \):
\[ E_{\text{max}} = 0.005 \, \text{H} \times 1000\pi \, \text{A/s} = 5\pi \, \text{V} \]

### Step 5: Relate the maximum emf to the given expression
We are given that:
\[ E_{\text{max}} = \frac{\pi}{x} \, \text{V} \]

Setting the two expressions for \( E_{\text{max}} \) equal:
\[ 5\pi = \frac{\pi}{x} \]

### Step 6: Solve for \( x \)
To solve for \( x \), we can divide both sides by \( \pi \) (as long as \( \pi \neq 0 \)):
\[ 5 = \frac{1}{x} \]

Taking the reciprocal gives:
\[ x = \frac{1}{5} \]

### Final Answer
The value of \( x \) is:
\[ x = 5 \]