A `12 Omega` resistance and an inductance of `0.05//pi` Hz with negligible resistance are connected in series. Across the ends of this circuit is connected a 130 V alternating voltage of frequency 50 Hz. Calculate the alternating current in the circuit and the potential difference across the resistance and that across the inductance.

Answer

To solve the given problem step by step, we will follow these calculations:

### Step 1: Calculate the Angular Frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
where \( f = 50 \) Hz.

Calculating ω:
\[ \omega = 2 \times \pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 2: Calculate the Inductive Reactance (XL)
The inductive reactance (XL) is given by the formula:
\[ X_L = \omega L \]
where \( L = \frac{0.05}{\pi} \) H.

Calculating \( X_L \):
\[ X_L = (100\pi) \times \left(\frac{0.05}{\pi}\right) = 100 \times 0.05 = 5 \, \Omega \]

### Step 3: Calculate the Impedance (Z)
The impedance (Z) in an LR circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
where \( R = 12 \, \Omega \) and \( X_L = 5 \, \Omega \).

Calculating Z:
\[ Z = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \Omega \]

### Step 4: Calculate the Alternating Current (I)
The alternating current (I) can be calculated using Ohm's law for AC circuits:
\[ I = \frac{V}{Z} \]
where \( V = 130 \, V \).

Calculating I:
\[ I = \frac{130}{13} = 10 \, A \]

### Step 5: Calculate the Potential Difference Across the Resistance (VR)
The potential difference across the resistance (VR) is given by:
\[ V_R = I \times R \]

Calculating \( V_R \):
\[ V_R = 10 \times 12 = 120 \, V \]

### Step 6: Calculate the Potential Difference Across the Inductance (VL)
The potential difference across the inductance (VL) is given by:
\[ V_L = I \times X_L \]

Calculating \( V_L \):
\[ V_L = 10 \times 5 = 50 \, V \]

### Summary of Results
- The alternating current in the circuit is \( 10 \, A \).
- The potential difference across the resistance is \( 120 \, V \).
- The potential difference across the inductance is \( 50 \, V \).

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