Embibe Experts Solutions for Chapter: Alternating Current, Exercise 1: Exercise

When a series $\mathrm{L}-\mathrm{C}-\mathrm{R} \mathrm{AC}$ circuit is brought into resonance, the current has a large value. Why?

Obtain an expression for resonant frequency ${\mathrm{f}}_0$ of a series $\mathrm{L}-\mathrm{C}-\mathrm{R}$ circuit.

An $\mathrm{AC}$ source of frequency $50 \mathrm{Hz}$ is connected to a $50 \mathrm{mH}$ inductor and a bulb. The bulb glows with some brightness. Calculate the capacitance of the capacitor to be connected in series with the circuit, so that the bulb glows with maximum brightness.

A $2 \mathrm{\mu F}$ capacitor, $100 \mathrm{\Omega }$ resistor and $8 \mathrm{H}$ inductor are connected in series with an $\mathrm{AC}$ source. What should be the frequency of this source for which the current drawn in the circuit is maximum$?$ If the peak value of emf of the source is $200 \mathrm{V}$ find for maximum current total impedance of the circuit.

Show that the average power dissipated in an $\mathrm{AC}$ circuit is, in general, given by

$\mathrm{P}={\mathrm{V}}_{\mathrm{rms}}\times {\mathrm{I}}_{\mathrm{rms}}\times \mathrm{R}/\mathrm{Z}$

where $\mathrm{R}$ is the resistance of the circuit (defined as real part of the complex impedance) and $\mathrm{Z}$ is the impedance.

A bulb of resistance $10 \mathrm{ohm}$, connected to an inductor of inductance $L$, is in series with an ac source marked $100\mathrm{V}$, $50\mathrm{Hz}$. If the phase angle between the voltage and angle is $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \mathrm{radian}$, calculate the value of $L$.

The emf and current of an a.c circuit containing $L,C \mathrm{and} R$ are given by $\epsilon ={\epsilon }_0\sin wt and I=I_0\sin \left(wt+\varphi \right)$. Show that the average power of the circuit is $P_{avg}={\epsilon }_v{I}_v\cos \varphi$, where ${\epsilon }_v$ and $I_v$ stand for virtual emf and virtual current respectively.

When a.c voltage supplied to LCR series circuit then damped oscillation is performed due to presence of