Question

The potential difference of the resistance, the capacitor and the inductor are $ 80\,V $ , $ 40\,V $ and $ 100\,V $ respectively in an LCR circuit. The power factor of the circuit is:
(A) $ 1.0 $
(B) $ 0.4 $
(C) $ 0.5 $
(D) $ 0.8 $

Answer 1

Hint
The power factor of the LCR circuit is determined by using the formula of the power factor of the LCR circuit. The power factor is depending on the potential difference of the resistance in the LCR circuit, the capacitor in the LCR circuit and the inductor in the LCR circuit.
Useful formula:
The power factor of the LCR circuit is given by,
 $ \Rightarrow p = \dfrac{{{V_R}}}{{\sqrt {V_R^2 + \left( {V_L^2 - V_C^2} \right)} }} $
Where, $ p $ is the power factor of the LCR circuit, $ {V_R} $ is the potential difference of the resistance, $ {V_L} $ is the potential difference of the inductor and $ {V_C} $ is the potential difference of the capacitor.

Complete step by step answer
Given that, The potential difference of the resistance is, $ {V_R} = 80\,V $ ,
The potential difference of the capacitor is, $ {V_C} = 40\,V $ ,
The potential difference of the inductor is, $ {V_L} = 100\,V $ .
Now, The power factor of the LCR circuit is given by,
 $ \Rightarrow p = \dfrac{{{V_R}}}{{\sqrt {V_R^2 + \left( {V_L^2 - V_C^2} \right)} }}\,....................\left( 1 \right) $
By substituting the potential difference of the resistance, the potential difference of the capacitor and the potential difference of the inductor in the above equation (1), then the above equation (1) is written as,
 $ \Rightarrow p = \dfrac{{80}}{{\sqrt {{{80}^2} + \left( {{{100}^2} - {{40}^2}} \right)} }} $
By squaring the terms in the above equation, then the above equation is written as,
 $ \Rightarrow p = \dfrac{{80}}{{\sqrt {6400 + \left( {10000 - 6400} \right)} }} $
By subtracting the terms in the above equation, then the above equation is written as,
 $ \Rightarrow p = \dfrac{{80}}{{\sqrt {6400 + 3600} }} $
By adding the terms in the above equation, then the above equation is written as,
 $ \Rightarrow p = \dfrac{{80}}{{\sqrt {10000} }} $
By taking the square root in the above equation, then the above equation is written as,
 $ \Rightarrow p = \dfrac{{80}}{{100}} $
By dividing the terms in the above equation, then the above equation is written as,
 $ \Rightarrow p = 0.8 $
Hence, the option (D) is the correct answer.

Note
The power factor is directly proportional to the potential difference of the resistance and inversely proportional to the square root of the sum of the individual square of the potential difference of the resistance and the difference of the potential difference of the inductor and potential difference of the capacitor.