LC series resonant circuit produces resonant frequency ' f '. If. ' L ' is tripled and ' C ' is increased by…

LC series resonant circuit produces resonant frequency ' f '. If. ' L ' is tripled and ' C ' is increased by 3C, the resonant frequency will be
  1. $\frac{\mathrm{f}}{3}$
  2. $\frac{\mathrm{f}}{2 \sqrt{3}}$
  3. 6 f
  4. $\frac{\mathrm{f}}{2 \sqrt{2}}$

Solution

$\begin{array}{ll} & \text { At resonance, } \mathrm{f}=\frac{1}{2 \pi \sqrt{\mathrm{LC}}} \Rightarrow \mathrm{f} \propto \frac{1}{\sqrt{\mathrm{LC}}} \\ \therefore \quad & \frac{\mathrm{f}^{\prime}}{\mathrm{f}}=\sqrt{\left(\frac{\mathrm{L}_1 \mathrm{C}_1}{\mathrm{~L}_2 \mathrm{C}_2}\right)}=\left(\frac{\mathrm{L} \times \mathrm{C}}{3 \mathrm{~L} \times 4 \mathrm{C}}\right)^{1 / 2}=\frac{1}{(12)^{1 / 2}} \\ \therefore \quad & \frac{\mathrm{f}^{\prime}}{\mathrm{f}}=\frac{1}{2 \sqrt{3}} \Rightarrow \mathrm{f}^{\prime}=\frac{\mathrm{f}}{2 \sqrt{3}}\end{array}$ ~

Asked in: MHT CET 2024 (09 May Shift 1)

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