Circuit problems with capacitors and parallel emf sources

I have quite a few questions so I’ve attached the associated images instead of making multiple threads. I’ll share my attempts/reasoning for the problems and please tell me if my logic is flawed or if I am doing something completely wrong.

Question 41: Bulb brightness is related to power output, correct? The bulbs must then have the same resistance and P = i2R so then the answer must be D since the total current is V/(R/2) = 2V/R and since the two bulbs are in parallel and equal, the each have current V/R, thus the power output is also V2/R in these bulbs. Any problems? Also, for P = iV, is this expression only true for the energy output from an emf source and P = i2R = V2/R only true for loads? Please correct me if I am mistaken. An explanation for why this is true would be greatly appreciated!

Question 43: P = 500 W and I = 15 A to ensure each bulb has minimum potential since they are in series and potential is added. (500/15) = potential for each bulb, 120 / (500/15) = 3.6 so must round down to 3. Therefore only 3 bulbs can be operated t full brightness from this line. Right?

Question 49: I was slightly confused on this one and believe my reasoning is fallacious. If I’m not mistaken, by connecting the two emf sources from cathode – cathode and anode – anode, this makes the total output emf ε2 – ε1 since ε2 > ε1. If it was not connected in this arrangement, but instead cathode-anode, would it be ε2 + ε1? Regardless, I found V = ε2 – ε1 because of this. Then current in the battery 1 is, I = ε1/r so P = iV = (ε2 – ε1)(ε1/r) but this is not the answer. I’m uncertain on how they arrived at their answer and am not sure if I’ve done anything wrong. If you could please tell mw what I did wrong and how to arrive at the proper conclusion, that would be much appreciated!

Question 62: I know: ## i(t) = ε/R (e−t/RC) when discharging and i(t) = 1 – ε/R (e−t/RC) when discharging. τa = ReqC is the discharging time constant where the loads are in series so Req = 2R. τb = RC/2 since Req = R/2 since the loads are in parallel. When doing τab = 4, but this is incorrect. Any help please? I’ve done question 63 and seem to have gotten it right, but knowing 62 will surely help in understanding it better.

Any help with these problems and additional advice or links to help understand circuit more comprehensively would be great! Also, is it possible to put two emf sources with different emf values in parallel? If so, will their emf values become equivalent very quickly? Is there a specific equation for this?

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