In a circuit, $$L,C$$  and $$R$$ are connected in series with an alternating voltage source of frequency $$f.$$ The current leads the voltage by $${45^ \circ }.$$ The value of $$C$$ is

Solution :
Phase difference between current and voltage in $$LCR$$  series circuit is given by
$$\tan \phi = \frac{{\omega L - \frac{1}{{\omega C}}}}{R}\,\,\left[ {_{\frac{1}{{\omega C}}\, = \,\,{\text{capacitive reactance}}}^{\omega L \,= \,\,{\text{inductive reactance}}}} \right]$$
$$\phi $$ being the angle by which the current leads the voltage.
Given, $$\phi = {45^ \circ }$$
$$\eqalign{ & \therefore \tan {45^ \circ } = \frac{{\omega L - \frac{1}{{\omega C}}}}{R} \cr & \Rightarrow 1 = \frac{{\omega L - \frac{1}{{\omega C}}}}{R} \cr & \Rightarrow R = \omega L - \frac{1}{{\omega C}} \cr} $$
$$\eqalign{ & \Rightarrow \omega C = \frac{1}{{\left( {\omega L - R} \right)}} \cr & \Rightarrow C = \frac{1}{{\omega \left( {\omega L - R} \right)}} \cr & = \frac{1}{{2\pi f\left( {2\pi fL - R} \right)}} \cr} $$