Energy in capacitor

[Mårten]

1. The problem statement, all variables and given/known data
A complete circuit with a battery U = 12 V, and two capacitors in series, C1 = 10 microF, C2 = 20 microF. How much energy is stored in C1?

2. Relevant equations
(1) W_c = 1/2 CV^2
(2) W_c = 1/2 Q^2/C
(3) C = Q/V
(4) C = 1/(C_1^{-1} + C_2^{-1})

3. The attempt at a solution
I tried to use equation (1) above with C=C1. I was thinking that the potential over each capacitor is 12 V, so I used V=12 V in equation (1). And I was supposing that the battery had already charged the capacitors, so they were fully charged. My W_c (=72 microjoule) then is not the same as in the key (=320 microjoule).

In the key, the solution starts with finding the substituted capacitor for the circuit, according to equation (4). Then they get Q from equation (3), to get the total charge in the substituted capacitor. And then, they use equation (2) with C=C1 to get W_c.

4. Questions
a) Why isn't my method working?
b) Isn't the voltage over each capacitor also 12 V?
c) How can equation (2) be used as they do, with C=C1? That equation should only apply for C (the substituted capacitor), because Q in that equation applies for the substituted capacitor, and not for C1. Or why could it apply to C1, when Q is the charge for the substituted capacitor and not for C1?

[Defennder]

I tried to use equation (1) above with C=C1. I was thinking that the potential over each capacitor is 12 V, so I used V=12 V in equation (1). And I was supposing that the battery had already charged the capacitors, so they were fully charged. My W_c (=72 microjoule) then is not the same as in the key (=320 microjoule).

In the key, the solution starts with finding the substituted capacitor for the circuit, according to equation (4). Then they get Q from equation (3), to get the total charge in the substituted capacitor. And then, they use equation (2) with C=C1 to get W_c.

4. Questions
a) Why isn't my method working?
b) Isn't the voltage over each capacitor also 12 V?
c) How can equation (2) be used as they do, with C=C1? That equation should only apply for C (the substituted capacitor), because Q in that equation applies for the substituted capacitor, and not for C1. Or why could it apply to C1, when Q is the charge for the substituted capacitor and not for C1?a)The potential across each capacitor is not 12 V. That is the potential difference across both capacitors. They're connected in series, not parallel.

b)See a)

c)Best way to do this question is as the answer key suggested: Finding equivalent capacitor. Note that the charge on capacitors in series are the same. The charge on an equivalent capacitor in series is the same as those on the individual series capacitors.

[Mårten]

a)The potential across each capacitor is not 12 V. That is the potential difference across both capacitors. They're connected in series, not parallel.

b)See a)

c)Best way to do this question is as the answer key suggested: Finding equivalent capacitor. Note that the charge on capacitors in series are the same. The charge on an equivalent capacitor in series is the same as those on the individual series capacitors.
Thank you very much - I think I got it!

Btw - how is it possible to have a current in this kind of circuit, when it's not complete? I mean, there are gaps in between the capacitor's plates, so the circuit cannot be complete.

[Defennder]

Well there isn't any current in steady state conditions, meaning to say once the capacitors are fully charged. But while it is still charging, electrons flow from the negative terminal to the uncharged plate, while electrons flow from the the other plate to the positive terminal, so while there isn't any movement of charges in the gaps, there is current elsewhere in the circuit.

[Mårten]

Ah...