LC Circuit Oscillations: Check My Work and Find Energy and Frequency Details

[davegillmour]

I'm not too confident in my work for this problem, so I was wondering if someone could check it over for me.

Consider a circuit with 4 elements, C1=100micro farads, C2=50micro farads, L1=20mH, and L2=10mH. At t=0, the capacitors are charged with Q=0.01 Coulomb. There is initially no current. (all 4 are connected in series)

a) What are the voltages across the capacitors?
V1=(1/C1)Q=(1/100micro farads)0.01= 100 Volts
V2=(1/50micro farads)0.01= 200 Volts
b)How much electrical and magnetic energy, respectively is stored in the circuit initially?
Ue=electric energy=q^2/(2C)
1/C=1/C1 +1/C2 C=33.3micro Farads
0.01^2/(2*33.3micro farads)=1.50 J
Ub=magnetic energy=(L*i^2)/2
i(initial)=0 so magnetic energy=0
c)What is the total inductance of the circuit.
L=L1+L2=20mH+10mH= 30mH
d)What is the total capacitance?
1/C=1/C1 + 1/C2 C=33.3micro farads
e)What is the frequency of oscillations in the circuit?
w=sqrt[1/(LC)] = sqrt[1/(30mH * 33.3micro farads)] = 1000 rad/s
f)What is q(t)? Make sure it satisfies q(0)=Q
q(t) = Qcos(wt + phi)
q(t) = (0.01)Cos(1000t) <---is the phase constant zero?
g)Compute i(t)= dq/dt
dq/dt= -10Sin(1000t)
h)Compute di/dt=d^2(q)/dt^2
di/dt= -10000Cos(1000t)

Thanks a lot to anyone who can do me this favor.

 

[lightgrav]

yes, the phase constant is zero since qharge on Cap's is maximum at t=0.
This is verified by the current being zero at t=0 (from the sine).

 

[kyle8921]

I am happy to review your work and provide feedback. Your calculations for the voltages across the capacitors and the total inductance and capacitance are correct. However, there are a few things to consider for the other parts of the problem:

b) The formula for electric energy stored in a capacitor is Ue=Q^2/(2C), not q^2/(2C). Remember that Q is the total charge on the capacitor, while q is the charge at a specific time t. So the correct calculation would be Ue=(0.01)^2/(2*33.3micro farads)=0.000005 J. For magnetic energy, it is true that at t=0, the initial current is 0, so there is no magnetic energy stored.

c) Your calculation for the total inductance is correct.

d) Your calculation for the total capacitance is incorrect. The correct calculation would be C=1/(1/C1 + 1/C2) = 33.3 micro farads.

e) Your calculation for the frequency of oscillations is also incorrect. The correct formula is w=1/sqrt(LC)=1/sqrt(30mH*33.3micro farads)= 816.5 rad/s.

f) q(t) is the charge on the capacitor at any given time t, so it cannot be a constant value of 0. The correct formula for q(t) would be q(t)=Qcos(wt + phi). The phase constant phi can be calculated using the initial conditions, in this case q(0)=Q=0.01 Coulomb. So phi=0.

g) Your calculation for i(t) is correct.

h) Your calculation for di/dt is also correct.

Overall, your work is mostly correct, but it's important to pay attention to the units and formulas being used. I hope this helps and good luck with your further studies!