A) \[\sqrt{\frac{3}{5}}\]
B) \[\sqrt{\frac{2}{5}}\]
C) \[\sqrt{\frac{1}{5}}\]
D) \[\sqrt{\frac{4}{5}}\]
Correct Answer: A
Solution :
At angular frequency w, the current in RC circuit is given by \[{{i}_{rms}}=\frac{{{V}_{rms}}}{\sqrt{{{R}^{2}}+{{\left( \frac{1}{\omega C} \right)}^{2}}}}\] ......(i) Also \[\frac{{{i}_{rms}}}{2}=\frac{{{V}_{rms}}}{\sqrt{{{R}^{2}}+{{\left( \frac{1}{\frac{\omega }{3}C} \right)}^{2}}}}=\frac{{{V}_{rms}}}{\sqrt{{{R}^{2}}+\frac{9}{{{\omega }^{2}}{{C}^{2}}}}}\] ......(ii) From equation (i) and (ii) we get \[3{{R}^{2}}=\frac{5}{{{\omega }^{2}}{{C}^{2}}}\Rightarrow \frac{\frac{1}{\omega C}}{R}=\sqrt{\frac{3}{5}}\]Þ \[\frac{{{X}_{C}}}{R}=\sqrt{\frac{3}{5}}\]
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