Question

In the shown $AC$ circuit $L$, $C$ and $V_o$ are fixed. What is the value of $\omega$ such that reading of hot wire ammeter shown is independent of 'R'?

• A. $\frac{1}{\sqrt{LC}}$
• B. $\sqrt{\frac{2}{LC}}$
• C. $\sqrt{\frac{3}{LC}}$
• D. $\frac{2}{\sqrt{LC}}$

Answer 1

The correct option is B $\sqrt{\frac{2}{LC}}$


$Z$ of top branch is $\sqrt{R^2+\frac{1}{{\omega }^2{C}^2}}$

Phase difference between currents of $RC$ branch and $L$ branch = ${90}^{\circ }+\theta$. where $tan\theta =\frac{1}{\omega CR}$

$i_{ammeter}^2={\left(\frac{V_o}{\omega L}\right)}^2+\frac{V_o^2}{Z^2}+\frac{2V_o^{2}cos(90+\theta )}{\omega LZ}$

$i_{ammeter}^2={\left(\frac{V_o}{\omega L}\right)}^2+\frac{V_o^2}{Z^2}-\frac{2V_o^{2}sin\theta }{\omega LZ}$

First term is independent of $R$.
So far $i_{ammeter}$ to be independent of $R$ second and third terms in the equation are to be equal.

$\Rightarrow \frac{V_o^2}{Z^2}=\frac{2V_o^{2}sin\theta }{\omega LZ}$
$\Rightarrow 2Zsin\theta =\omega L\Rightarrow 2\frac{1}{\omega C}=\omega L\Rightarrow \omega =\sqrt{\frac{2}{LC}}$