A coil of inductance `4//pi` H is joined in series with a resistance of `30 Omega`. Calculate the current flowing in the circuit when connected to a.c. mains of 200 V and frequency 50 Hz.

Answer

To solve the problem step by step, we will follow the outlined procedure to find the current flowing in the circuit.

### Step 1: Identify the given values
- Inductance (L) = \( \frac{4}{\pi} \) H
- Resistance (R) = 30 Ω
- Voltage (E) = 200 V
- Frequency (f) = 50 Hz

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 3: Calculate the inductive reactance (XL)
The inductive reactance (XL) is calculated using the formula:
\[ X_L = \omega L \]
Substituting the values of ω and L:
\[ X_L = (100\pi) \times \left(\frac{4}{\pi}\right) \]
The π cancels out:
\[ X_L = 100 \times 4 = 400 \, \Omega \]

### Step 4: Calculate the total impedance (Z)
The total impedance (Z) in an LR circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values of R and XL:
\[ Z = \sqrt{30^2 + 400^2} \]
Calculating the squares:
\[ Z = \sqrt{900 + 160000} = \sqrt{160900} \]
Calculating the square root:
\[ Z \approx 401.1 \, \Omega \]

### Step 5: Calculate the current (I)
Using Ohm's law for AC circuits, the current (I) can be calculated as:
\[ I = \frac{E}{Z} \]
Substituting the values of E and Z:
\[ I = \frac{200}{401.1} \]
Calculating the current:
\[ I \approx 0.4986 \, \text{A} \]

### Final Answer
The current flowing in the circuit is approximately \( 0.4986 \, \text{A} \).

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