An inductor of 0.5 mH, a capacitor of 200 $$\mu$$F and a resistor of 2 $$\Omega$$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the frequency of ac source will be ______ $$\times 10^2$$ Hz.

We need to find the frequency at which the current is in phase with the emf in an RLC series circuit (resonance condition). Current is in phase with emf when the circuit is at resonance. At resonance: $$\omega_0 = \frac{1}{\sqrt{LC}}$$ and $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$.

Substituting the given values: $$L = 0.5$$ mH $$= 0.5 \times 10^{-3}$$ H, $$C = 200\ \mu$$F $$= 200 \times 10^{-6}$$ F gives $$LC = 0.5 \times 10^{-3} \times 200 \times 10^{-6} = 100 \times 10^{-9} = 10^{-7}$$ and $$\sqrt{LC} = \sqrt{10^{-7}} = 10^{-3.5} = \frac{10^{-3}}{\sqrt{10}}$$. Then $$f_0 = \frac{1}{2\pi \times \frac{10^{-3}}{\sqrt{10}}} = \frac{\sqrt{10}}{2\pi \times 10^{-3}}$$, which equals $$= \frac{3.162}{6.2832 \times 10^{-3}} = \frac{3.162}{6.2832} \times 10^3$$, yielding $$= 0.5033 \times 10^3 \approx 503.3 \text{ Hz}$$ or approximately $$\approx 5 \times 10^2 \text{ Hz}$$. The answer is 5 $$\times 10^2$$ Hz.