A) \[\sqrt{0.6}\]
B) \[\sqrt{3}\]
C) \[\sqrt{2}\]
D) \[\sqrt{6}\]
Correct Answer: A
Solution :
: In the first case, Capacitive reactance, \[{{X}_{C}}=\frac{1}{\omega C}\] Impedance of the circuit, \[Z=\sqrt{{{R}^{2}}+X_{C}^{2}}\] Current in the circuit \[I=\frac{V}{R}=\frac{V}{\sqrt{{{R}^{2}}+X_{C}^{2}}}\] ??.(i) In the second case, \[\omega =\frac{\omega }{3}\] \[\therefore \] Capacitive reactance, \[X_{C}^{}=\frac{1}{\omega C}=\frac{1}{\frac{\omega }{3}C}=\frac{3}{\omega C}=3{{X}_{C}}\] Impedance of the circuit \[Z=\sqrt{{{R}^{2}}+X_{C}^{2}}=\sqrt{{{R}^{2}}+9X_{C}^{2}}\] Current in the circuit \[\frac{1}{2}=\frac{V}{Z}=\frac{V}{\sqrt{{{R}^{2}}+9X_{C}^{2}}}\] Divide (ii) by (i), we get \[\frac{1}{2}=\frac{\sqrt{{{R}^{2}}+X_{C}^{2}}}{{{R}^{2}}+9X_{C}^{2}}\] Squaring both sides, we get \[\frac{1}{4}=\frac{{{R}^{2}}+X_{C}^{2}}{{{R}^{2}}+9X_{C}^{2}}\] \[{{R}^{2}}+9X_{C}^{2}=4{{R}^{2}}+4X_{C}^{2}\] \[5X_{C}^{2}=3{{R}^{2}}\] or \[\frac{{{X}_{C}}}{R}=\sqrt{\frac{3}{5}}=\sqrt{0.6}\]
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