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. $\frac{1}{2}$
• E. 4

Answer 1

Answer: (A)

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 $Z_2=\sqrt{R^2+\frac{1}{4{\pi }^2(2f^2)C^2}}$
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.