Question : Three students X,Y and Z performed an experiment for studying the variation of alternating currents with angular frequency in a series LCR circuit and obtained the graphs shown below. They all used AC sources of the same RMS value and Inductances of the same value.
What can we (qualitatively) conclude about the (i) capacitive values (ii) resistance used by them? In which case will the quality factor be maximum?
What can we conclude about the nature of the impedance of the setup at the frequency ω₀?
Doubt by Bhavya
Solution :
(i) We know,
ω₀ = 1/√(LC)
SBS
ω₀² = 1/LC
C = 1/ω₀²L
Since, ω₀ and L are same for all three students so C should also be same for all three cases.
CX=CY=CZ
(ii) At resonance:
I₀ = V₀/R
All sources have the same RMS voltage, so a higher peak current means lower resistance.
From the graph:
IX> IY > IZ
Therefore,
RX < RY < RZ
We know
Q = ω₀L/R
Since ω₀ and L are the same, Q is maximum when R is minimum. Therefore, quality factor is maximum for X.
At resonance,
XL = XC
Therefore, the inductive and capacitive reactances cancel each other. Hence,
Z = R
So, at ω₀ the circuit has purely resistive impedance.
