Medium
Suppose the frequency of the source in the previous example can be varied.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.
(N/A) The frequency at which resonance occurs is given by the formula $\omega_{0} = \frac{1}{\sqrt{LC}}$.
Substituting the values $L = 25.48 \times 10^{-3} \, H$ and $C = 796 \times 10^{-6} \, F$:
$\omega_{0} = \frac{1}{\sqrt{25.48 \times 10^{-3} \times 796 \times 10^{-6}}} = 222.1 \, rad/s$.
The resonant frequency in Hertz is $v_{r} = \frac{\omega_{0}}{2\pi} = \frac{222.1}{2 \times 3.14} \approx 35.4 \, Hz$.
$(b)$ At resonance,the inductive reactance equals the capacitive reactance $(X_{L} = X_{C})$,so the impedance $Z$ is equal to the resistance $R$:
$Z = R = 3 \, \Omega$.
The rms current at resonance is $I = \frac{V_{rms}}{Z} = \frac{V_{peak} / \sqrt{2}}{R} = \frac{283 / 1.414}{3} \approx 66.7 \, A$.
The power dissipated at resonance is $P = I^{2}R = (66.7)^{2} \times 3 \approx 13.35 \, kW$.
Substituting the values $L = 25.48 \times 10^{-3} \, H$ and $C = 796 \times 10^{-6} \, F$:
$\omega_{0} = \frac{1}{\sqrt{25.48 \times 10^{-3} \times 796 \times 10^{-6}}} = 222.1 \, rad/s$.
The resonant frequency in Hertz is $v_{r} = \frac{\omega_{0}}{2\pi} = \frac{222.1}{2 \times 3.14} \approx 35.4 \, Hz$.
$(b)$ At resonance,the inductive reactance equals the capacitive reactance $(X_{L} = X_{C})$,so the impedance $Z$ is equal to the resistance $R$:
$Z = R = 3 \, \Omega$.
The rms current at resonance is $I = \frac{V_{rms}}{Z} = \frac{V_{peak} / \sqrt{2}}{R} = \frac{283 / 1.414}{3} \approx 66.7 \, A$.
The power dissipated at resonance is $P = I^{2}R = (66.7)^{2} \times 3 \approx 13.35 \, kW$.
Explore More
Similar Questions
$A$ series $LCR$ circuit is connected across a source of alternating emf of changing frequency and resonates at frequency $f_0$. Keeping capacitance constant,if the inductance $(L)$ is increased by $\sqrt{3}$ times and resistance is increased $(R)$ by $1.4$ times,the resonant frequency now is
An alternating current is flowing through a series $L-C-R$ circuit. It is found that the current reaches a value of $1 \ mA$ at both $200 \ Hz$ and $800 \ Hz$ frequency. What is the resonance frequency of the circuit (in $Hz$)?
In an $L-C-R$ series circuit,the value of only capacitance $C$ is varied. The resulting variation of resonance frequency $f_0$ as a function of $C$ can be represented as
$A$ telegraph line of length $100 \, km$ has a capacity of $0.01 \, \mu F/km$ and it carries an alternating current at $0.5 \, kHz$. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is . . . . . . $mH$. (Take $\pi = \sqrt{10}$)
$A$ series $LCR$ circuit is connected to a $45 \sin (\omega t) \text{ V}$ source. The resonant angular frequency of the circuit is $10^5 \text{ rad s}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8 \times 10^4 \text{ rad s}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L = 50 \text{ mH}$, match each entry in List-$I$ with an appropriate value from List-$II$ and choose the correct option.
List-$I$ List-$II$ $(P)$ $I_0$ in $\text{mA}$ $(1)$ $44.4$ $(Q)$ The quality factor of the circuit $(2)$ $18$ $(R)$ The bandwidth of the circuit in $\text{rad s}^{-1}$ $(3)$ $400$ $(S)$ The peak power dissipated at resonance in $\text{Watt}$ $(4)$ $2250$ $(5)$ $500$
| List-$I$ | List-$II$ |
| $(P)$ $I_0$ in $\text{mA}$ | $(1)$ $44.4$ |
| $(Q)$ The quality factor of the circuit | $(2)$ $18$ |
| $(R)$ The bandwidth of the circuit in $\text{rad s}^{-1}$ | $(3)$ $400$ |
| $(S)$ The peak power dissipated at resonance in $\text{Watt}$ | $(4)$ $2250$ |
| $(5)$ $500$ |
MediumIIT 2023
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo