A telegraph line of length $$100$$ km has a capacity of $$0.01$$ $$\mu$$F km$$^{-1}$$ and it carries an alternating current at $$0.5$$ kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is ______ mH. (If $$\pi = \sqrt{10}$$)

We need to find the inductance required for minimum impedance (resonance condition) in a telegraph line. The line length is $$100$$ km and the capacitance per km is $$0.01$$ $$\mu$$F/km, so the total capacitance is $$C = 100 \times 0.01 = 1$$ $$\mu$$F $$= 10^{-6}$$ F. The frequency is $$f = 0.5$$ kHz $$= 500$$ Hz and $$\pi = \sqrt{10}$$.

The resonance condition for minimum impedance in a series LC circuit states that:

$$ \omega L = \frac{1}{\omega C} $$

This gives:

$$ L = \frac{1}{\omega^2 C} $$

Since $$\omega = 2\pi f = 2\sqrt{10} \times 500 = 1000\sqrt{10}$$, we have $$\omega^2 = (1000\sqrt{10})^2 = 1000000 \times 10 = 10^7$$. Substituting into the expression for $$L$$ with $$C = 10^{-6}$$ F yields:

$$ L = \frac{1}{10^7 \times 10^{-6}} = \frac{1}{10} = 0.1 \text{ H} = 100 \text{ mH} $$

Hence, the value of inductance required is 100 mH.