What is electrical resonance in a series LCR circuit? Derive the expression for the resonant frequency of a series LCR circuit.

Q: What is electrical resonance in a series LCR circuit? Derive the expression for the resonant frequency of a series LCR circuit.

Electrical Resonance in Series LCR Circuit:\nElectrical resonance is a phenomenon in a series LCR circuit in which, at a particular frequency of the applied alternating voltage, the inductive reactance ($XL$) becomes equal to the capacitive reactance ($XC$). At this specific frequency, the total impedance of the circuit becomes minimum and equal to the resistance ($Z = R$), resulting in a maximum flow of alternating current through the circuit. Derivation of Resonant Frequency ($fr$):\nThe impedance of a series LCR circuit is given by: $$Z = \sqrt{R^2 + (XL - XC)^2}$$ \nAt resonance, $XL = XC$. \nSince $XL = \omegar L$ and $XC = \frac{1}{\omegar C}$: $$\omegar L = \frac{1}{\omegar C}$$ $$\omegar^2 = \frac{1}{LC}$$ $$\omegar = \frac{1}{\sqrt{LC}}$$ \nSince angular frequency $\omegar = 2\pi fr$, where $fr$ is the resonant frequency: $$2\pi fr = \frac{1}{\sqrt{LC}}$$ $$fr = \frac{1}{2\pi\sqrt{LC}}$$ \nThis expression gives the resonant frequency of a series LCR circuit.

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Electrical Resonance in Series LCR Circuit:
Electrical resonance is a phenomenon in a series LCR circuit in which, at a particular frequency of the applied alternating voltage, the inductive reactance ($XL$) becomes equal to the capacitive reactance ($XC$). At this specific frequency, the total impedance of the circuit becomes minimum and equal to the resistance ($Z = R$), resulting in a maximum flow of alternating current through the circuit. Derivation of Resonant Frequency ($fr$):
The impedance of a series LCR circuit is given by: $$Z = \sqrt{R^2 + (XL - XC)^2}$$ 
At resonance, $XL = XC$. 
Since $XL = \omegar L$ and $XC = \frac{1}{\omegar C}$: $$\omegar L = \frac{1}{\omegar C}$$ $$\omegar^2 = \frac{1}{LC}$$ $$\omegar = \frac{1}{\sqrt{LC}}$$ 
Since angular frequency $\omegar = 2\pi fr$, where $fr$ is the resonant frequency: $$2\pi fr = \frac{1}{\sqrt{LC}}$$ $$fr = \frac{1}{2\pi\sqrt{LC}}$$ 
This expression gives the resonant frequency of a series LCR circuit.

CBSE · Class 12 · Physics · Alternating CurrentWhat is electrical resonance in a series LCR circuit? Derive the expression for the resonant frequency of a series LCR circuit.