In the given circuit, the rms value of current (\( I_{\text{rms}} \)) through the resistor \( R \) is:
In an AC circuit, the impedance \( Z \) is given by \( Z = \sqrt{R^2 + (X_L -X_C)^2} \), where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance. The rms current can be calculated using \( I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} \).
Step 1: Problem Definition The circuit comprises a resistor \( R = 100\Omega \), inductive reactance \( X_L = 200\Omega \), and capacitive reactance \( X_C = 100\Omega \). The applied rms voltage is \( V_{\text{rms}} = 200\sqrt{2}V \). The objective is to determine the rms current flowing through the resistor. Step 2: Impedance Calculation The circuit's impedance (\( Z \)) is calculated using the formula: \[ Z = \sqrt{R^2 + (X_L -X_C)^2}. \] Plugging in the provided values: \[ Z = \sqrt{100^2 + (200 -100)^2} = \sqrt{10000 + 10000} = \sqrt{20000} = 100\sqrt{2}\Omega. \] Step 3: RMS Current Calculation The rms current (\( I_{\text{rms}} \)) is computed using Ohm's law for AC circuits: \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}. \] Substituting the determined values: \[ I_{\text{rms}} = \frac{200\sqrt{2}}{100\sqrt{2}} = 2A. \] Step 4: Option Selection The computed rms current is \(2A\), which aligns with option (A). Final Answer: The rms current through the resistor is 2A.