Question

The impedance of a $RC$ circuit is ${{Z}_{1}}$ for frequency $f$ and ${{Z}_{2}}$ for frequency $2f$ . Then, ${{Z}_{1}}/{{Z}_{2}}$ is

• A. between 1 and 2
• B. 2
• C. 2 between $\frac{1}{2}$ and 1
• D. $\frac{1}{2}$

Answer 1

Answer: (A)

between 1 and 2

Answer 2

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.