A) between 1 and 2
B) 2
C) 2 between\[\frac{1}{2}\]and 1
D) \[\frac{1}{2}\]
E) 4
Correct Answer: A
Solution :
The impedance of R-C circuit for frequency\[{{f}_{1}}\]is \[{{Z}_{1}}=\sqrt{{{R}^{2}}+\frac{1}{4{{\pi }^{2}}{{f}^{2}}{{C}^{2}}}}\] The impedance of JR-C circuit for frequency\[2f\]is \[{{Z}_{2}}=\sqrt{{{R}^{2}}+\frac{1}{4{{\pi }^{2}}(2{{f}^{2}}){{C}^{2}}}}\] Or \[{{Z}_{2}}=\sqrt{{{R}^{2}}+\frac{1}{16{{\pi }^{2}}{{f}^{2}}{{C}^{2}}}}\] Then \[\frac{Z_{1}^{2}}{Z_{2}^{2}}=\frac{{{R}^{2}}+\frac{1}{4{{\pi }^{2}}{{f}^{2}}{{C}^{2}}}}{{{R}^{2}}+\frac{1}{16{{\pi }^{2}}{{f}^{2}}{{C}^{2}}}}\] Or \[\frac{Z_{1}^{2}}{Z_{2}^{2}}=\frac{1+\frac{1}{4{{\pi }^{2}}{{f}^{2}}{{C}^{2}}{{R}^{2}}}}{1+\frac{1}{16{{\pi }^{2}}{{f}^{2}}{{R}^{2}}{{C}^{2}}}}\] value is greater than 1. Hence,\[\frac{{{Z}_{1}}}{{{Z}_{2}}}=\]lies between 1 and 2.
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