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Alternating Current — Physics STD 12 Science — Question

Gujarat 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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