Question

Consider the given arrangement of a resistor, inductor and capacitor connected with an AC source as shown in figure. Choose the correct option(s)

• A. Overall arrangement behaves as capacitive.
• B. Phase difference between voltage and current from the source is $ta{n}^{-1}\frac{1}{2}$.
• C. Equivalent impedance of the arrangement is $\frac{4R}{5}$.
• D. Equivalent reactance of the arrangement is $\frac{2R}{5}$.

Answer 1

Answer: (B), (D)

The correct options are
B Phase difference between voltage and current from the source is $ta{n}^{-1}\frac{1}{2}$.
D Equivalent reactance of the arrangement is $\frac{2R}{5}$.


$I_1=\frac{E_0}{R}\sin (\omega t+\frac{\pi }{4})$

$I_2=\frac{E_0}{R}\sin (\omega t+\frac{\pi }{4}-\frac{\pi }{2})$

$I_3=\frac{E_0}{2R}\sin (\omega t+\frac{\pi }{4}+\frac{\pi }{2})$



Since current lags voltage, the overall circuit is inductive.
Option 1 is wrong.

From the figure,
$\tan \varphi =\frac{1}{2}$.
Option 2 is correct.

$I=\frac{\sqrt{5}{E}_0}{2R}\sin (\omega t+\frac{\pi }{4}-\varphi )$

$Z=\sqrt{R_1^2+X^2}=\frac{2R}{\sqrt{5}}$

Option 3 is wrong.

$\Rightarrow R_1^2+X^2=\frac{4R^2}{5}$

$\tan \varphi =\frac{X}{R_1}=\frac{1}{2}$

$\Rightarrow R_1=2X$

$\Rightarrow 4X^2+X^2=5X^2=\frac{4R^2}{5}$

$\Rightarrow X^2=\frac{4R^2}{25}$

$\Rightarrow X=\frac{2R}{5}$

Option 4 is correct.