Lcr Series Resonance

NCERT Class 12, Chapter 7 (Alternating Current) shows that a series LCR circuit resonates when the inductive and capacitive reactances become equal, so that $\omega L=\frac{1}{\omega C}$, giving the resonant angular frequency ${\omega }_0=\frac{1}{\sqrt{LC}}$. At resonance the impedance is purely resistive and minimum, so the current is maximum. Substituting $L=2\text{H}$ and $C=8\times 10^{-6}\text{F}$: $LC=2\times 8\times 10^{-6}=1.6\times 10^{-5}$, so $\sqrt{LC}=4\times 10^{-3}\text{s}$, and ${\omega }_0=\frac{1}{4\times 10^{-3}}=250\text{rad/s}$. The option 125 rad/s halves the result by a square-root error. The option 500 rad/s doubles it. The option 62.5 rad/s divides by an extra factor of four. As a plausibility check, the units of $\frac{1}{\sqrt{\text{H}\cdot \text{F}}}$ reduce to rad/s, and resonance frequencies of this order are typical for such component values used in tuning circuits. An important insight is that the resonant frequency depends only on the inductance and capacitance, and is completely independent of the resistance in the circuit; the resistance instead controls how sharp the resonance peak is and how much current flows at resonance. At this special frequency the voltage across the inductor and the voltage across the capacitor are equal in magnitude but opposite in phase, so they cancel exactly, leaving the source to see only the resistance. This is why series resonance is sometimes called acceptor resonance, since the circuit most readily accepts current at precisely this frequency.