A 0.21 H inductor and a `12 Omega ` resistance are connected in series to a `220 V, 50 Hz ac source.
The current in the circuit is

Options

A. `(220)/(sqrt(4400))A`
B. `(22)/(3sqrt(5)) A`
C. `(220)/(sqrt(4600))A`
D. `(22)/(5sqrt(3)) A`

Explanation

To solve the problem of finding the current in a circuit with a 0.21 H inductor and a 12 Ω resistor connected in series to a 220 V, 50 Hz AC source, we can follow these steps:

### Step 1: Calculate the Angular Frequency (ω)
The angular frequency (ω) is calculated using the formula:
\[ \omega = 2\pi f \]
where \( f \) is the frequency in hertz (Hz).

Given \( f = 50 \, \text{Hz} \):
\[ \omega = 2 \pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 2: Calculate the Inductive Reactance (XL)
The inductive reactance (XL) can be calculated using the formula:
\[ X_L = \omega L \]
where \( L \) is the inductance in henries (H).

Given \( L = 0.21 \, \text{H} \):
\[ X_L = 100\pi \times 0.21 = 21\pi \, \Omega \]

### Step 3: Calculate the Impedance (Z)
The impedance (Z) in a series circuit with resistance (R) and inductive reactance (XL) is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]

Given \( R = 12 \, \Omega \):
\[ Z = \sqrt{(12)^2 + (21\pi)^2} \]
Calculating \( (21\pi)^2 \):
\[ (21\pi)^2 = 441\pi^2 \]
Now substituting back:
\[ Z = \sqrt{144 + 441\pi^2} \]

### Step 4: Calculate the Current (I)
The current (I) in the circuit can be calculated using Ohm's Law for AC circuits:
\[ I = \frac{V}{Z} \]
where \( V \) is the voltage.

Given \( V = 220 \, \text{V} \):
\[ I = \frac{220}{\sqrt{144 + 441\pi^2}} \]

### Step 5: Simplify and Calculate
To find a numerical value for \( Z \):
1. Calculate \( 441\pi^2 \):
- \( \pi^2 \approx 9.87 \)
- \( 441\pi^2 \approx 4359.27 \)
2. Now, add \( 144 \):
- \( 144 + 4359.27 \approx 4503.27 \)
3. Take the square root:
- \( Z \approx \sqrt{4503.27} \approx 67.08 \, \Omega \)

Now substituting \( Z \) back into the current formula:
\[ I \approx \frac{220}{67.08} \approx 3.28 \, \text{A} \]

### Final Answer
The current in the circuit is approximately \( 3.28 \, \text{A} \).

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