A series LCR circuit is connected to a 45 sin (ωt) volt source. The resonant angular frequency of the circuit is 105 rad/sec and the current amplitude at resonance is I0. When the angular frequency of the source is ω = 8 x 104 rad/sec, the current amplitude in the circuit is 0.05 I0. If m = 50 mH, match each entry in the list - I with an approximate value from list - II and choose the option.
List - I List - II (P) I0 in mA (1) 44.4 (Q) The quality factor of the circuit (2) 18 (R) The bandwidth of the circuit in rad/sec (3) 400 (S) The peak power dissipated at resonance in watt (4) 2250 (5) 500
A series LCR circuit is connected to a 45 sin (ωt) volt source. The resonant angular frequency of the circuit is 105 rad/sec and the current amplitude at resonance is I0. When the angular frequency of the source is ω = 8 x 104 rad/sec, the current amplitude in the circuit is 0.05 I0. If m = 50 mH, match each entry in the list - I with an approximate value from list - II and choose the option.
| List - I | List - II | ||
| (P) | I0 in mA | (1) | 44.4 |
| (Q) | The quality factor of the circuit | (2) | 18 |
| (R) | The bandwidth of the circuit in rad/sec | (3) | 400 |
| (S) | The peak power dissipated at resonance in watt | (4) | 2250 |
| (5) | 500 |
P Q R S 4 3 2 1 P Q R S 2 1 3 4 P Q R S 3 1 2 5 P Q R S 3 1 4 2
The Correct Option is D
Solution and Explanation
The correct option is: (D)
\(E=45\,sin\omega t\)
\(\omega_rL=\frac{1}{\omega_rC}\)
\(\omega_r^2=\frac{1}{LC}\)
\((10^5)^2=\frac{1}{50\times10^{-3}\times C}\)
\(10^{10}=\frac{1}{5\times10^{-2}C}\Rightarrow c=2\times10^{-9}F\)
at \(\omega=8\times10^4\,rad/sec\)
\(X_L=\omega_L=8\times10^4\times50\times10^{-3}=4000\Omega\)
\(X_C=\frac{1}{\omega_c}=\frac{1}{8\times10^4\times2\times10^{-9}}=6250\Omega\)
\(X=X_C-X_L=2250\Omega\)
Also,
\(0.05I_0=\frac{45}{Z}\)
\(0.5\times \frac{45}{R}=\frac{45}{Z}\)
\(Z=\frac{R}{0.05}\)
\(\sqrt{R^2+x^2}=20R\)
\(R^2+x^2=400R^2\)
\(\Rightarrow R=112.6\Omega\,\,\,\,(as\, x=2250\Omega)\)
\((P)I_0=\frac{45}{112.6}A=\frac{45\times1000}{112.6}mA\approx400mA\)
\((Q)Q_{factor}=\frac{\Omega_r\times L}{R}=\frac{10^5\times50\times10^{-3}}{112.6}=44.4\)
\((R)\) B and width = \(\frac{R}{L}=2250\,rad/sec\)
\((S)\) Peek power at resonance = \(\frac{(45)^2}{R}=\frac{45^2}{112.6}\approx 18\omega\)
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Concepts Used:
Oscillations
Oscillation is a process of repeating variations of any quantity or measure from its equilibrium value in time . Another definition of oscillation is a periodic variation of a matter between two values or about its central value.
The term vibration is used to describe the mechanical oscillations of an object. However, oscillations also occur in dynamic systems or more accurately in every field of science. Even our heartbeats also creates oscillations. Meanwhile, objects that move to and fro from its equilibrium position are known as oscillators.
Read More: Simple Harmonic Motion
Oscillation- Examples
The tides in the sea and the movement of a simple pendulum of the clock are some of the most common examples of oscillations. Some of examples of oscillations are vibrations caused by the guitar strings or the other instruments having strings are also and etc. The movements caused by oscillations are known as oscillating movements. For example, oscillating movements in a sine wave or a spring when it moves up and down.
The maximum distance covered while taking oscillations is known as the amplitude. The time taken to complete one cycle is known as the time period of the oscillation. The number of oscillating cycles completed in one second is referred to as the frequency which is the reciprocal of the time period.