Question

The LCR circuit shown below is driven by an ideal AC voltage source.

At frequency $f=\frac{1}{2\pi \sqrt{LC}}$, the correct statement is

• A. The current through $R$ is zero
• B. The current through $R$ is infinite
• C. The current through $R$ depends on the value of $L$ and $C$
• D. The current through $R$ depends only on the value of $R$ and not $L$ and $C$

Answer 1

Answer: (A)

The $AC$ circuit is shown in the following figure,

Frequency of $AC$ source is given as
$f=\frac{1}{2\pi \sqrt{LC}}$
Clearly, it is resonant frequency.
Hence, $X_L=X_C$
In this case, voltages developed across $L$ and $C$ are equal in magnitude but opposite in pollaries.
$\therefore Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$
$=\sqrt{R^2+0^2}$
$Z=R$
$\therefore I=\frac{V}{Z}=\frac{V}{R}$
Hence, the current through $R$ depends only on the value of $R$ and not $L$ and $C$.