Question

The applied voltage in an alternating circuit is 220 V. If \( R = 8 \, \Omega \) and \( X_L = X_C = 6 \, \Omega \), then find the root mean square value of voltage and impedance of the circuit.

Hint:

Impedance \(Z = \sqrt{R^2 + (X_L - X_C)^2}\). If \(X_L = X_C\), impedance equals the resistance \(R\). RMS voltage \(V_{\text{RMS}} = \frac{V_{\text{applied}}}{\sqrt{2}}\).

Answer 1

In an alternating current (AC) circuit, the impedance \(Z\) is given by the relation: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] Where: - \(R\) is the resistance of the circuit, - \(X_L\) is the inductive reactance, - \(X_C\) is the capacitive reactance. Here, we are given that: - \(R = 8 \, \Omega\), - \(X_L = X_C = 6 \, \Omega\). Since \(X_L = X_C\), their difference is zero. Therefore, the impedance simplifies to: \[ Z = \sqrt{R^2} = R \] Substituting the given value of \(R\): \[ Z = 8 \, \Omega \] Now, the root mean square (RMS) value of the voltage is given by the formula: \[ V_{\text{RMS}} = \frac{V_{\text{applied}}}{\sqrt{2}} \] Where \(V_{\text{applied}} = 220 \, \text{V}\). So, \[ V_{\text{RMS}} = \frac{220}{\sqrt{2}} \approx 155.56 \, \text{V} \] Thus, the impedance of the circuit is \(8 \, \Omega\), and the RMS value of the voltage is approximately \(155.56 \, \text{V}\).