When 100 V DC is applied across a solenoid, a steady current of 1 A flows in it. when 100V AC is applied across the same solenoid, the current drops to 0.5 A. If the frequency of the Ad source is `150// sqrt3pi Hz`, the impedance and inductance of the solenoid are

Options

A. `200 Omega` and `1//3 H`
B. `100 Omega` and `1//16 H`
C. `200 Omega` and `1.0 H`
D. `1100 Omega` and `3//11 H`

Explanation

To solve the problem step by step, we will calculate the resistance, impedance, inductive reactance, and finally the inductance of the solenoid.

### Step 1: Calculate the Resistance (R)
When a 100 V DC is applied across the solenoid, the steady current is 1 A. We can use Ohm's law to find the resistance.

\[ R = \frac{V}{I} \]

Where:
- \( V = 100 \, \text{V} \)
- \( I = 1 \, \text{A} \)

Substituting the values:

\[ R = \frac{100 \, \text{V}}{1 \, \text{A}} = 100 \, \Omega \]

### Step 2: Calculate the Impedance (Z)
When a 100 V AC is applied across the same solenoid, the current drops to 0.5 A. We can again use Ohm's law to find the impedance.

\[ Z = \frac{V_{AC}}{I_{AC}} \]

Where:
- \( V_{AC} = 100 \, \text{V} \)
- \( I_{AC} = 0.5 \, \text{A} \)

Substituting the values:

\[ Z = \frac{100 \, \text{V}}{0.5 \, \text{A}} = 200 \, \Omega \]

### Step 3: Calculate the Inductive Reactance (X_L)
The inductive reactance can be calculated using the formula:

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

Substituting the values we found:

\[ X_L = \sqrt{(200 \, \Omega)^2 - (100 \, \Omega)^2} \]

Calculating:

\[ X_L = \sqrt{40000 - 10000} = \sqrt{30000} = 100\sqrt{3} \, \Omega \]

### Step 4: Calculate the Inductance (L)
The inductance can be calculated using the formula:

\[ L = \frac{X_L}{2\pi f} \]

Where:
- \( f = \frac{150}{\sqrt{3}\pi} \, \text{Hz} \)

Substituting \( X_L \) and \( f \):

\[ L = \frac{100\sqrt{3}}{2\pi \left(\frac{150}{\sqrt{3}\pi}\right)} \]

Simplifying:

\[ L = \frac{100\sqrt{3}}{2 \cdot 150} = \frac{100\sqrt{3}}{300} = \frac{100}{100} = 1 \, \text{H} \]

### Final Results
- Impedance \( Z = 200 \, \Omega \)
- Inductance \( L = 1 \, \text{H} \)

### Summary
The impedance of the solenoid is \( 200 \, \Omega \) and the inductance is \( 1 \, \text{H} \).

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