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Series LCR Circuit and Phase
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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 -In a series LCR circuit, $\tan\phi = \dfrac{X_C - X_L}{R}$, where $\phi$ is the angle by which the current leads the voltage.
The current leads the voltage, so the circuit is capacitive: $X_C \gt X_L$.
Given $\phi = 45^\circ$: $\tan 45^\circ = 1 = \dfrac{X_C - X_L}{R}$
Here $X_C - X_L = R$
$\Rightarrow \dfrac{1}{2\pi fC} = R + 2\pi fL$
$\Rightarrow C = \dfrac{1}{2\pi f(2\pi fL + R)}$