Consider an L-C circuit with capacitance , inductance , and no voltage source, a

Question

Consider an L-C circuit with capacitance , inductance , and no voltage source, as shown in the figure (Figure 1)

. As a function of time, the charge on the capacitor is and the current through the inductor is . Assume that the circuit has no resistance and that at one time the capacitor was charged.

Part A

Part B

Part C

Consider an L - C circuit with capacitance , inductance , and no voltage source, as shown in the figure (Figure 1) . As a function of time, the charge on the capacitor is and the current through the inductor is . Assume that the circuit has no resistance and that at one time the capacitor was charged. Part A As a function of time, what is the energy U_L(t) stored in the inductor? Express your answer in terms of L and I(t). Part B As a function of time, what is the energy U_C(t)stored within the capacitor? Express your answer in terms of C and Q(t). Part C What is the total energy U_total stored in the circuit? Express your answer in terms of the maximum current I_0 and inductance L.

Explanation / Answer

a)The energy in an inductor is
UL(t)=(1/2)LI(t)2

b)The energy in a capacitor is

UC(t)=(1/2)Q(t)2/C

c)I(t)=Iocost

Without an external voltage source, the frequency will be the resonant frequency of the capacitor and inductor

so =1/(LC)

VC(t)=Vocos(t-/2)

Vo =I/C

Q(t)=CVc(t) =C(Io/C)cos(t-/2) =(Io/)sint

The total energy is then

Utotal = UL(t) + UC(t) =(1/2)LIo2cos2(t)+(1/2C)(Io2/2)sin2(t)

Utotal =(1/2)LIo2cos2(t)+(1/2C)(LCIo2)sin2(t)

Utotal =(1/2)LIo2(cos2(t)+sin2(t))

Utotal =(1/2)LIo2

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