The current in a series RLC circuit leads the generator voltage by φ = 30°. The circuit, containing an inductor L = 400 mH and a resistor R = 50 Ω, is driven by a generator operating at ω = 100 rad/s with a maximum emf of 10 V. The capacitance is unknown.
The first question regarding this prompt is to find the capacitance, which is 145e-6F.
This is the question I am stumped on:
In terms of the maximum energy UC,max stored in the capacitor and the angle φ by which the current leads the generator voltage, the energy UC stored in the capacitor when the time is t = π/2ω is:
The correct answer is UC = UC,max sin^2φ
The Attempt at a Solution
I've drawn my phasor diagrams, but I don't understand the correct answer. Is the voltage across the capacitor the component of the voltage across the generator pointing in the direction of X_c, the capacitor's impedance? If so, I get how a sin^2(phi) term would be used, because that would be the V^2 part of U = 1/2*cV^2, but I don't see how that would give me UC,max *sin^2(phi).
I will appreciate any help!