Question

A series LCR circuit consists of \( R = 80\,\Omega \), \( X_L = 100\,\Omega \), and \( X_C = 40\,\Omega \). The input voltage is \( 2500 \cos(100\pi t) \) V. The amplitude of current in the circuit is

• A. 25 A
• B. 50 A
• C. 75 A
• D. 100 A

Hint:

Always use the peak voltage (\( V_0 \)) to find the amplitude of current. If the question asks for the "current" without specifying amplitude, it usually refers to the RMS value, where \( I_{rms} = I_0 / \sqrt{2} \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
In an AC circuit, the amplitude of current (\( I_0 \)) is found by dividing the peak voltage (\( V_0 \)) by the total impedance (\( Z \)) of the circuit.

Step 2: Key Formula or Approach
1. Impedance \( Z = \sqrt{R^2 + (X_L - X_C)^2} \) 2. Peak current \( I_0 = \frac{V_0}{Z} \)

Step 3: Detailed Explanation
1. Identify Peak Voltage: From the equation \( V = 2500 \cos(100\pi t) \), the amplitude \( V_0 = 2500 \) V. 2. Calculate Impedance (\( Z \)): - \( R = 80\,\Omega \) - \( X_L - X_C = 100 - 40 = 60\,\Omega \) - \( Z = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100\,\Omega \). 3. Calculate Current Amplitude (\( I_0 \)): - \( I_0 = \frac{2500}{100} = 25 \) A.

Step 4: Final Answer
The amplitude of current in the circuit is 25 A.