Question

A box $\text{P}$ and a coil $\text{Q}$ are connected in series with an $AC$ source of variable frequency. The EMF of the source is constant at $10\text{V}$. Box $\text{P}$ contains a capacitance of $1\mu \text{F}$ in series with a resistance of $32\Omega$. Coil $\text{Q}$ has a self inductance of $4.9\text{mH}$ and a resistance of $68\Omega$. The frequency is adjusted so that the maximum current flows in $\text{P}$ and $\text{Q}$. If $Z_P$ and $Z_Q$ represents the impedance of box $\text{P}$ and $\text{Q}$ respectively, then

• A. $Z_P=77\Omega$
• B. $Z_P=38\Omega$
• C. $Z_Q=49\Omega$
• D. $Z_Q=98\Omega$

Answer 1

Answer: (A), (D)

$Z_Q=98\Omega$

The current flowing is the $LCR$ circuit is given as

$i=\frac{E}{\sqrt{[R^2+{(\omega L-\frac{1}{\omega C})}^2]}}$

The current in maximum at resonance, when

$(\omega L-\frac{1}{\omega C})=0$

$\Rightarrow {\omega }^2=\frac{1}{\sqrt{LC}}$

$\Rightarrow \omega =\frac{1}{\sqrt{(4.9\times {10}^{-3})\left({10}^{-6}\right)}}$

$\Rightarrow \omega =\frac{{10}^5}{7}\text{rad/s}$

The impedance of $\text{P}$ is given as

$Z_P=\sqrt{(R_1^2+X_C^2)}=\sqrt{[R_1^2+{\left(\frac{1}{\omega C}\right)}^2]}$

$Z_P={\left[\right(32{)}^2+{(\frac{7}{{10}^5}\times \frac{1}{{10}^{-6}})}^2]}^{\frac{1}{2}}$

$\therefore Z_P=77\Omega$

The impedance of $\text{Q}$ is given as

$Z_Q=\sqrt{[R_2^2+{\omega }^2{L}^2]}$

$Z_Q=\sqrt{\left[\right(68{)}^2+{(\frac{{10}^5}{7}\times 4.9\times {10}^{-3})}^2]}$

$\therefore Z_Q=98\Omega$

Hence, options $\left(A\right)$ and $\left(D\right)$ are correct.