Maximum energy stored in a capacitor? RLC circuit

[Luke Cohen]

Homework Statement

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

Homework Equations

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!

 

[Luke Cohen]

This is just a normal box series RLC circuit with a battery, resistor, capacitor, and inductor in series. Sorry for not including!

 

[ehild]

Y

Homework Statement

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

Homework Equations

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!

You have to work with time dependence of the current and voltages. Assuming the generator voltage is V_g=V_g0sin(ωt),
what is the time dependence of the current and the capacitor voltage?

 

[NascentOxygen]

This is just a normal box series RLC circuit with a battery, resistor, capacitor, and inductor in series. Sorry for not including!

There is no escaping this: you will need to show, and then use, the phasor diagram of current and voltages.