1. Given data

• Length of telegraph line: $l = 100\ \text{km}$
• Capacity per km: $0.01\ \mu\text{F/km}$
• Frequency: $0.5$ kilocycle/s $= 0.5\times 10^3 = 500\ \text{Hz}$

We need the inductance to be introduced in series so that the impedance is minimum.

2. Condition for minimum impedance

For a series $L-C$ combination, impedance is minimum at resonance:

$X_L = X_C$

So,

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

Hence,

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

3. Total capacitance of the line

Capacity per km is $0.01\ \mu\text{F}$. For $100\ \text{km}$,

$C = 100 \times 0.01\ \mu\text{F} = 1\ \mu\text{F} = 10^{-6}\ \text{F}$

4. Angular frequency

$\omega = 2\pi f = 2\pi(500) = 1000\pi$

Given $\pi = \sqrt{10}$,

$\omega = 1000\sqrt{10}$

Thus,

$\omega^2 = (1000\sqrt{10})^2 = 10^6 \times 10 = 10^7$

5. Calculate inductance

$L = \frac{1}{\omega^2 C} = \frac{1}{10^7 \times 10^{-6}} = \frac{1}{10} = 0.1\ \text{H}$

Convert into mH:

$0.1\ \text{H} = 100\ \text{mH}$

6. Final answer

The required inductance is

$\boxed{100\ \text{mH}}$