An inductor of 10 mH, capacitor of 0.1\ μ F and a resistor of 100\...

The power factor of an LCR series circuit is given by $\cos \phi = \dfrac{R}{Z}$.

Given $\cos \phi = 0.5$ and $R = 100 \ \Omega$.

Substituting the values, we get:

$0.5 = \dfrac{100}{Z} \Rightarrow Z = 200 \ \Omega$

The impedance $Z$ of the circuit is also given by:

$Z = \sqrt{R^2 + (X_L - X_C)^2}$

Squaring both sides and substituting the values of $Z$ and $R$:

$(200)^2 = (100)^2 + (X_L - X_C)^2$

$40000 = 10000 + (X_L - X_C)^2$

$(X_L - X_C)^2 = 30000$

$|X_L - X_C| = 100\sqrt{3} \ \Omega$

The difference in inductive and capacitive reactance is given as $\sqrt{3}\alpha \ \Omega$.

Comparing the two expressions:

$\sqrt{3}\alpha = 100\sqrt{3}$

$\alpha = 100$

Answer: $100$