Question

For series $\text{RLC}$, circuit graph between current and frequency of voltage source is shown in figure.

$\text{Assertion}:$- Nature of circuit in region $\left(1\right)$ is capacitive while in region $\left(2\right)$ is inductive.

$\text{Reason}:$- Power factor of circuit in region $\left(1\right)$ is negative and in region $\left(2\right)$ is positive.

• A. Both $\text{Assertion}$ & $\text{Reason}$ are true & the $\text{Reason}$ is a correct explanation of the $\text{Assertion}$.
• B. Both $\text{Assertion}$ & $\text{Reason}$ are true, but $\text{Reason}$ is a correct explanation of the $\text{Assertion}$.
• C. $\text{Assertion}$ is true, but the $\text{Reason}$ is false.
• D. Both $\text{Assertion}$ & $\text{Reason}$ are false.

Answer 1

The correct option is C $\text{Assertion}$ is true, but the $\text{Reason}$ is false.

For $\text{LCR}$ circuit,

$i_0=\frac{V_0}{Z}=\frac{V_0}{\sqrt{R^2+(X_L-X_C{)}^2}}$

So $i_0$ will be maximum when,

$X_L=X_C$.

$2\pi fL=\frac{1}{2\pi fC}$

$\Rightarrow f=\frac{1}{2\pi \sqrt{LC}}=f_r$

Now as we see,

$X_L\propto f$, and $X_C\propto \frac{1}{f}$

So for $f<f_r$,

$X_L<X_C\to$ nature is Capacitive

And for $f>f_r$,

$X_L>X_C\to$ nature is inductive.

Now phase factor $=\cos \varphi$
where $\varphi$ is phase difference between $i_{net}$ & $V_{net}$.

For $X_L>X_C$:-

For $X_L<X_C$:-

So phase factor is positive for both cases.

Hence, option $\left(\text{C}\right)$ is correct.

Note: <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> For Power factor: $PF=\cos \varphi =\frac{R}{Z}$ and both $R$ and $Z$ are positive.