Question

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of 20 μF is .............. V.

Answer 1

Correct Answer: 50

The inductive reactance is:

\[ X_L = \omega L = 100 \times 1 = 100 \, \Omega. \]

The capacitive reactance is:

\[ X_C = \frac{1}{\omega C} = \frac{1}{100 \times 20 \times 10^{-6}} = 500 \, \Omega. \]

The total impedance is:

\[ Z = \sqrt{(X_L - X_C)^2 + R^2} = \sqrt{(100 - 500)^2 + 300^2}. \]

Simplifying:

\[ Z = \sqrt{(-400)^2 + 300^2} = \sqrt{160000 + 90000} = 500 \, \Omega. \]

The rms current is:

\[ i_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{50}{500} = 0.1 \, \text{A}. \]

The rms voltage across the capacitor is:
\[ V_{\text{rms, capacitor}} = X_C \cdot i_{\text{rms}} = 500 \times 0.1 = 50 \, \text{V}. \]

Answer 2

Step 1: Given data
The circuit parameters are:
Inductance \(L = 1\,\text{H}\)
Capacitance \(C = 20\,\mu\text{F} = 20 \times 10^{-6}\,\text{F}\)
Resistance \(R = 300\,\Omega\)
AC supply: \(V = 50\sqrt{2} \sin(100t)\,\text{V}\).
Hence, angular frequency \(\omega = 100\,\text{rad/s}\).

Step 2: Calculate the reactances
Inductive reactance: \[ X_L = \omega L = 100 \times 1 = 100\,\Omega. \] Capacitive reactance: \[ X_C = \frac{1}{\omega C} = \frac{1}{100 \times 20 \times 10^{-6}} = \frac{1}{2 \times 10^{-3}} = 500\,\Omega. \] Hence, the net reactance is \[ X = X_L - X_C = 100 - 500 = -400\,\Omega. \] The negative sign indicates the circuit is capacitive.

Step 3: Find the impedance of the circuit
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{(300)^2 + (-400)^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500\,\Omega. \]

Step 4: Find RMS current
Given the supply \(V = 50\sqrt{2}\sin(100t)\), the RMS value is \(V_{\text{rms}} = 50\,\text{V}\).
So, \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{50}{500} = 0.1\,\text{A}. \]

Step 5: Find RMS potential difference across the capacitor
\[ V_C = I_{\text{rms}} \times X_C = 0.1 \times 500 = 50\,\text{V}. \]

Step 6: Final answer
The RMS potential difference across the 20 μF capacitor is:

50 V