Question

In the given circuit, the rms value of current (\( I_{\text{rms}} \)) through the resistor \( R \) is: 

• A. \(2A\)
• B. \(\frac{1}{2}A\)
• C. \(20A\)
• D. \(2\sqrt{2}A\)

Hint:

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 \( I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} \).

Answer 1

The Correct Option is A

Step 1: Understanding the Problem 
We are given an AC circuit with a resistor \( R = 100\Omega \), inductive reactance \( X_L = 200\Omega \), and capacitive reactance \( X_C = 100\Omega \). The rms voltage \( V_{\text{rms}} = 200\sqrt{2}V \). We need to find the rms current through the resistor. 
Step 2: Calculating the Impedance of the Circuit 
The impedance (\( Z \)) of the circuit is given by: \[ Z = \sqrt{R^2 + (X_L -X_C)^2}. \] 
Substituting the given values: \[ Z = \sqrt{100^2 + (200 -100)^2} = \sqrt{10000 + 10000} = \sqrt{20000} = 100\sqrt{2}\Omega. \] 
Step 3: Calculating the rms Current 
The rms current (\( I_{\text{rms}} \)) is given by: \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}. \] 
Substituting the values: \[ I_{\text{rms}} = \frac{200\sqrt{2}}{100\sqrt{2}} = 2A. \] 
Step 4: Matching with the Options 
The calculated rms current is \(2A\), which corresponds to option (A). Final Answer: The rms value of current through the resistor is 2A.