Question 99 of 190

An AC source is connected to an inductance of $100mH$, a capacitance of $100\mu F$ and a resistance of $120\Omega$ as shown in figure. The time in which the resistance having a thermal capacity $2JC-1^{\circ}$ will get heated by $16^{\circ}C$ is _____ $s$.

Your answerCheck answer

Show answer and solution▾

Answer: 15

From the diagram given,

Angular frequency $\omega =100rad{s}^{-1}$ and ${V}_{rms}=\dfrac{{V}_{0}}{\sqrt{2}}=\dfrac{20\sqrt{2}}{\sqrt{2}}=20V$.
Now, ${X}_{L}=\omega L=100\times 100\times {10}^{-3}=10\Omega$ and ${X}_{C}=\dfrac{1}{\omega C}=\dfrac{1}{100\times 100\times {10}^{-6}}=100\Omega$
$\left|\left({X}_{L}-{X}_{C}\right)\right|=\left|10-100\right|=90\Omega$
The impedance of the circuit,
$Z=\sqrt{{\left({X}_{L}-{X}_{C}\right)}^{2}+{R}^{2}}=\sqrt{{\left(90\right)}^{2}+{\left(120\right)}^{2}}=150\Omega$
The current in the circuit,
${i}_{mrs}=\dfrac{{V}_{rms}}{Z}=\left(\dfrac{20}{150}\right)A$
Power factor $\cos \phi =\dfrac{R}{Z}$
Now
$\left({V}_{rms}{i}_{rms}\cos \phi \right)\Delta t=ms\left(\Delta T\right) \Rightarrow \Delta t=\dfrac{ms\left(\Delta T\right)}{{V}_{rms}\times \dfrac{{V}_{rms}}{Z}\times \dfrac{R}{Z}}=\dfrac{ms\left(\Delta T\right){Z}^{2}}{{{V}_{rms}}^{2}\times R}=\dfrac{2\times 16\times 22500}{400\times 120}=15s$
$\Rightarrow \Delta t=15s$