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Alternating Current — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsAlternating Current3 Marks
Question
An $AC$ voltage $V = V_m$ is applied across a:
  1. Series $RC$ circuit in which capacitive reactance is $‘a\ ’$ times the resistance in the circuit.
  2. Series $RL$ circuit in which inductive reactance is $‘b\ ’$ times the resistance in the circuit.
Find the value of power factor of the circuit in each case.
✓

Answer

Power factor $\cos\varphi=\frac{\text{R}}{\text{Z}},$ when $\text{Z}=\sqrt{\text{R}^2+\text{X}^2}$
  1. $\text{X = X}_{\text{C}} = \text{aR},$
$\therefore\text{Z}=\sqrt{\text{R}^2+(\text{aR})^2}=\text{R}\sqrt{1+\text{a}^2}$
$\therefore\cos\varphi=\frac{\text{R}}{\text{R}\sqrt{1+\text{a}^2}}=\frac{1}{\sqrt{1+\text{a}^2}}$
  1. $\text{X = X}_{\text{L}}=\text{bR},$
$\therefore\text{Z}=\sqrt{\text{R}^2+(\text{bR})^2}=\text{R}\sqrt{1+\text{b}^2}$
$\therefore\cos\varphi=\frac{\text{R}}{\text{R}\sqrt{1+\text{b}^2}}=\frac{1}{\sqrt{1+\text{b}^2}}$

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