Question

Given below are two statements :  
Statement I : When the frequency of an a.c. source in a series LCR circuit increases, the current in the circuit first increases, attains a maximum value and then decreases.  
Statement II : In a series LCR circuit, the value of power factor at resonance is one. 
In the light of given statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are true.
• B. Both Statement I and Statement II are false.
• C. Statement I is correct but Statement II is false.
• D. Statement I is incorrect but Statement II is true.

Answer 1

The Correct Option is A

To analyze the given statements, we will break down each one and examine the underlying physics principles.

1. Statement I Analysis:

   The statement claims that in a series LCR circuit, with increasing frequency of an AC source, the current first increases, reaches a maximum value, and then decreases. This behavior is characteristic of resonance in an AC circuit. In a series LCR circuit, the condition for resonance occurs at a particular frequency called the resonant frequency, given by:

   f_r = \frac{1}{2\pi\sqrt{LC}}

   At this resonant frequency, the impedance of the circuit is minimum and equal to the resistance R (since the inductive and capacitive reactances cancel each other). Consequently, the current is maximum at this frequency. As the frequency deviates from the resonant frequency, the impedance increases, which results in a decrease in current. Hence, this statement is correct.

2. Statement II Analysis:

   The statement asserts that the power factor in a series LCR circuit at resonance is one. The power factor is defined as:

   \text{Power Factor} = \cos\phi = \frac{R}{Z}

   where Z is the impedance of the circuit, and \phi is the phase difference between the current and voltage. At resonance, the impedance Z is purely resistive and equal to R, leading to a zero phase difference (\phi = 0). Therefore, the power factor becomes:

   \cos\phi = \frac{R}{R} = 1

   This means that all the power supplied is effectively used, with no reactive power. Thus, Statement II is also correct.

3. Conclusion:

   Given that both Statement I and Statement II are rooted in well-established principles of AC circuit resonance behavior and power factor analysis, the correct conclusion is:

Correct Answer: Both Statement I and Statement II are true.