For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is $$2.5 \text{ nF}$$. If resistance of $$200 \Omega$$ and $$100 \text{ mH}$$ inductor is being used in the given circuit. The frequency of ac source is ___________ $$\times 10^3 \text{ Hz}$$. (given $$\pi^2 = 10$$)

At resonance in series LCR,

$$f=\frac{1}{2\pi\sqrt{LC}}$$

Given,

$$L=100mH=0.1H$$

$$C=2.5\text{ nF}=2.5\times10^{-9}F$$

So,

$$f=\frac{1}{2\pi\sqrt{(0.1)(2.5\times10^{-9})}}$$

$$=\frac{1}{2\pi\sqrt{2.5\times10^{-10}}}$$

$$\sqrt{2.5\times10^{-10}}=\frac{5\times10^{-5}}{\sqrt{10}}$$

Thus,

$$f=\frac{1}{2\pi\cdot\frac{5\times10^{-5}}{\sqrt{10}}}$$

Given

$$\pi^2=10\Rightarrow\sqrt{10}=\pi$$

Hence,

$$f=\frac{1}{2\pi\cdot\frac{5\times10^{-5}}{\pi}}$$

$$=\frac{1}{10\times10^{-5}}$$

$$=10^4\text{ Hz}$$

$$=10\times10^3\text{ Hz}$$