Charge on C1 at t1 in LC Circuit

In summary, LC circuit has two capacitors in series with the inductor in between. The capacitors have values of 409 μF and 294 μF respectively. The inductance is 397 mH. At time t=0, the current through the inductor has its maximum value IL(0) = 64 mA and it has the direction shown (CW) from capacitors on left to inductor on right. The current through the inductor has its maximum value IL(t) at different times due to the cosine function it follows. When the current energy (magnitude!) is maximum the capacitor energy is minimum, and vice-versa. The total max potential energy in the inductor is 8.
  • #1
theoB
3
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LC circut: Charge on a capacitor at a given time

Homework Statement




I don’t have a pic but a simple loop circuit is constructes with 2 capacitors c1 and c2 in series, after the capacitors in an inductor also in series, and the wire continues back into c1. The values for the capacitors are: C1 = 409 μF and C2 = 294 μF. The inductance is L = 397 mH. At time t =0, the current through the inductor has its maximum value IL(0) = 64 mA and it has the direction shown (CW) from capacitors on left to inductor on right.

What is Q1(t1), the charge on the capacitor C1 at time t = t1 = 30.2 ms?

Homework Equations



Q(t)=Qmax*cos(Omega*t+phase angle)

this equation gives the charge on a capacitor as a function of time

The Attempt at a Solution



Using the equivalent capacitance I found the the angular frequency omega was 121.352 rad/sec which is correct, however when trying to find q max I get a bit lost, they tell me that the current through the inductor has its maximum value IL(0) = 64 mA, so I tried to use energy equations

U capacitor =1/2 c*v^2 and U inductor= ½ L*i^2

I found that the total max potential energy in the inductor is 8.135e-4
I don’t exactly know where to go from here. I can solve for the V and thus Q in the equivalent capacitor but that doesn’t exactly help me since I need the Q through only c1
Am I on the right track or did I go wrong somewhere?
 
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  • #2
You're doing okay. A couple of concepts may help you.

The LC circuit operates by swapping energy between the capacitor and inductor. When the current energy (magnitude!) is maximum the capacitor energy is minimum, and vice-versa. You have given the expressions for the energy for both inductor and capacitance. So you should be able to find the maximum voltage that the net capacitance will see during its cycle. Perhaps call it Vmax.

From the given time t=0, the current in the inductor is going to follow a cosine function with amplitude Imax (64mA as given in the problem statement) and angular frequency that you've calculated. The Capacitance voltage will be zero when the inductor current is maximum (energy consideration). Which basic trig function starts its cycle at zero? (note, pay attention to which direction the voltage should be headed initially).

The second concept is that of the capacitive voltage divider. Given two capacitors in series and a total voltage V impressed across them, what's the individual voltages across each capacitor?
 
  • #3
So for the phase shift I could use Pi or change it to a sin function and disregard the phase shift. The second concept is where I think I am hung up, I know the max voltage across the equivalent capacitor, can I simply relate the total voltage and equivalent capacitance to a specific voltage over a single capacitor? thanks.
 
  • #4
I realized my obvious error, which was that if I knew the equivalent capacitance as well as the voltage over the equivalent capacitor I could easily calculate the charge… which would be the SAME on both capacitors in series. Thank you for your time it is really appreciated
 
  • #5




Thank you for sharing your problem with us. It seems like you are on the right track in your approach to solving for the charge on capacitor C1 at time t1. The equation you provided, Q(t)=Qmax*cos(Omega*t+phase angle), is correct for finding the charge on a capacitor as a function of time.

In order to solve for Qmax, you will need to use the conservation of energy principle. As you correctly pointed out, the total maximum potential energy in the inductor is 8.135e-4. This energy will be shared between the capacitors and the inductor in the circuit. In order to find the specific charge on capacitor C1, you will need to use the energy equations for both the inductor and the equivalent capacitance.

Since the inductor and the equivalent capacitance are in series, the total energy will be the sum of the energies in both components. You can then use this total energy to solve for the charge on capacitor C1 at time t1.

I hope this helps guide you in the right direction. Remember to always check your units and make sure they are consistent in your calculations. Good luck!
 

1. What is a charge on C1 at t1 in an LC Circuit?

The charge on C1 at t1 in an LC Circuit refers to the amount of electrical charge stored on the capacitor C1 at a specific time t1. It is a measure of the capacitor's ability to store and release electrical energy.

2. How is the charge on C1 at t1 calculated in an LC Circuit?

The charge on C1 at t1 can be calculated using the formula Q = CV, where Q is the charge (in coulombs), C is the capacitance (in farads), and V is the voltage (in volts) across the capacitor at time t1.

3. Why is the charge on C1 at t1 important in an LC Circuit?

The charge on C1 at t1 is important because it affects the behavior of the circuit. The amount of charge stored on the capacitor determines the voltage and current in the circuit, which can impact the overall performance and stability of the circuit.

4. How does the charge on C1 at t1 change over time in an LC Circuit?

In an LC Circuit, the charge on C1 at t1 will initially increase as the capacitor charges up, and then decrease as the energy is released back into the circuit. This process repeats itself, leading to oscillations in the charge on C1 at t1 over time.

5. Can the charge on C1 at t1 be controlled in an LC Circuit?

Yes, the charge on C1 at t1 can be controlled by adjusting the capacitance or voltage in the circuit. By changing these parameters, the amount of charge stored on the capacitor can be increased or decreased, leading to changes in the behavior of the circuit.

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