An inductor of inductance L Henry is connected to an AC supply of 12 V . The frequency of source is 50 Hz and current in the inductor is 0.5 A. When the same inductor is connected across a direct current supply of 12 V it draws the current of 1.5 A . Find the value of L .

Answer

To find the value of the inductance \( L \) in the given problem, we will follow these steps:

### Step 1: Determine the resistance \( R \) using the DC supply
When the inductor is connected to a DC supply of 12 V, it draws a current of 1.5 A. Using Ohm's law, we can find the resistance \( R \):

\[ R = \frac{V}{I} = \frac{12 \, \text{V}}{1.5 \, \text{A}} = 8 \, \Omega \]

### Step 2: Determine the impedance \( Z \) using the AC supply
When the inductor is connected to an AC supply of 12 V and the current is 0.5 A, we can find the impedance \( Z \):

\[ Z = \frac{V}{I} = \frac{12 \, \text{V}}{0.5 \, \text{A}} = 24 \, \Omega \]

### Step 3: Relate impedance \( Z \), resistance \( R \), and inductive reactance \( X_L \)
For an LR circuit, the impedance \( Z \) is given by:

\[ Z = \sqrt{R^2 + X_L^2} \]

We already have \( R = 8 \, \Omega \) and \( Z = 24 \, \Omega \). We can substitute these values into the equation:

\[ 24^2 = 8^2 + X_L^2 \]

Calculating this gives:

\[ 576 = 64 + X_L^2 \]

### Step 4: Solve for inductive reactance \( X_L \)
Rearranging the equation to solve for \( X_L^2 \):

\[ X_L^2 = 576 - 64 = 512 \]

Taking the square root:

\[ X_L = \sqrt{512} \approx 22.63 \, \Omega \]

### Step 5: Relate inductive reactance \( X_L \) to inductance \( L \)
The inductive reactance \( X_L \) is related to the inductance \( L \) and the angular frequency \( \omega \):

\[ X_L = L \omega \]

Where \( \omega = 2 \pi f \) and \( f = 50 \, \text{Hz} \):

\[ \omega = 2 \pi \times 50 \approx 314 \, \text{rad/s} \]

### Step 6: Solve for \( L \)
Now we can substitute \( X_L \) and \( \omega \) into the equation to find \( L \):

\[ L = \frac{X_L}{\omega} = \frac{22.63}{314} \approx 0.072 \, \text{H} \text{ or } 72 \, \text{mH} \]

### Final Answer
The value of the inductance \( L \) is approximately \( 0.072 \, \text{H} \) or \( 72 \, \text{mH} \).

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