An AC source is connected to an inductance of 100 mH, a capacitance of 100μ F and a resistance of 120Ω as shown in figure. The time in which the…

From the diagram given,

Angular frequency $\omega =100\mathrm{rad}{s}^{-1}$ω=100rads−1 and ${V}_{rms}=\frac{{V}_{0}}{\sqrt{2}}=\frac{20\sqrt{2}}{\sqrt{2}}=20V$Vrms​=2​V0​​=2​202​​=20V.

Now, ${X}_{L}=\omega L=100\times 100\times {10}^{-3}=10\Omega$XL​=ωL=100×100×10−3=10Ω and ${X}_{C}=\frac{1}{\omega C}=\frac{1}{100\times 100\times {10}^{-6}}=100\Omega$XC​=ωC1​=100×100×10−61​=100Ω

$$|({X}_{L}-{X}_{C})|=|10-100|=90\Omega$$∣(XL​−XC​)∣=∣10−100∣=90Ω

The impedance of the circuit,

$$Z=\sqrt{{({X}_{L}-{X}_{C})}^{2}+{R}^{2}}=\sqrt{{(90)}^{2}+{(120)}^{2}}=150\Omega$$Z=(XL​−XC​)2+R2​=(90)2+(120)2​=150Ω

The current in the circuit,

$${i}_{\mathrm{mrs}}=\frac{{V}_{\mathrm{rms}}}{Z}=(\frac{20}{150})A$$imrs​=ZVrms​​=(15020​)A

Power factor $\mathrm{cos}\phi =\frac{R}{Z}$cosϕ=ZR​

Now

$$({V}_{rms}{i}_{rms}\mathrm{cos}\phi )\Delta t=ms(\Delta T) \Rightarrow \Delta t=\frac{ms(\Delta T)}{{V}_{rms}\times \frac{{V}_{rms}}{Z}\times \frac{R}{Z}}=\frac{ms(\Delta T){Z}^{2}}{{{V}_{rms}}^{2}\times R}=\frac{2\times 16\times 22500}{400\times 120}=15s$$(Vrms​irms​cosϕ)Δt=ms(ΔT)⇒Δt=Vrms​×ZVrms​​×ZR​ms(ΔT)​=Vrms​2×Rms(ΔT)Z2​=400×1202×16×22500​=15s

$$\Rightarrow \Delta t=15s$$⇒Δt=15s