Basic capacitor & ideal gas questions

Both of these questions are from this paper.

Question 4
One of the correct answers is obviously "The charges on the plates are equal and opposite". However, I don't understand why the other answer is "In the first 10 s the capacitor gains 180uJ of energy". According to the formula sheet, the energy stored in a capacitor is 0.5CV^2, which means the energy stored in the capacitor when it is fully charged would be 0.5*10*10^-6*6^2=180uJ. However, surely this would be the energy stored when the capacitor is fully charged, not after 10 seconds? Why would the capacitor become fully charged after 10 seconds? I firstly thought it's impossible for a capacitor to become 100% fully charged and anyway, how would you even go about working this question out? I know the equations for how current and charge change with respect to time but I don't know how to relate this to energy?

Both of these questions are from this paper.

Question 4
One of the correct answers is obviously "The charges on the plates are equal and opposite". However, I don't understand why the other answer is "In the first 10 s the capacitor gains 180uJ of energy". According to the formula sheet, the energy stored in a capacitor is 0.5CV^2, which means the energy stored in the capacitor when it is fully charged would be 0.5*10*10^-6*6^2=180uJ. However, surely this would be the energy stored when the capacitor is fully charged, not after 10 seconds? Why would the capacitor become fully charged after 10 seconds? I firstly thought it's impossible for a capacitor to become 100% fully charged and anyway, how would you even go about working this question out? I know the equations for how current and charge change with respect to time but I don't know how to relate this to energy?

The voltage across the capacitor and hence the charge on it, is governed by the relationship:

$V_{C_{(t)}} = V_S (1 - e^{(\frac{-t}{CR})})$

$Q = CV$

$Q_{(t)} = CV_S (1 - e^{(\frac{-t}{CR})})$

Notice that the voltage follows an exponential law increase whose asymptote is the supply voltage.

The time constant product of C and R means the capacitor will achieve >99% of the infinite-time maximum possible charge in 5CR time periods.

Since CR = 1 second, then by 5 seconds, the capacitor has achieved >99% full charge and for practical purposes, it is deemed to have achieved "full charge" although strictly speaking it can only do this as t approaches infinity.

Energy stored is

$\frac{1}{2}CV^2$

where the above voltage follows the exponential relationship at time t seconds.

$V = V_{C_{(t)}}$

Both of these questions are from this paper.

Question 4
One of the correct answers is obviously "The charges on the plates are equal and opposite". However, I don't understand why the other answer is "In the first 10 s the capacitor gains 180uJ of energy". According to the formula sheet, the energy stored in a capacitor is 0.5CV^2, which means the energy stored in the capacitor when it is fully charged would be 0.5*10*10^-6*6^2=180uJ. However, surely this would be the energy stored when the capacitor is fully charged, not after 10 seconds? Why would the capacitor become fully charged after 10 seconds? I firstly thought it's impossible for a capacitor to become 100% fully charged and anyway, how would you even go about working this question out? I know the equations for how current and charge change with respect to time but I don't know how to relate this to energy?

Question 11.c.ii.
Substituting pV=NkT into the equation for energy density, I get Energy Density = 3NkT/V (or Eint/V=3NkT/V) and since N, k and V are constants (as the volume in the boiler is constant), the energy density should be directly proportional to temperature. However, the mark scheme shows a different trend (increasing gradient from 0). Why is this?

Well the capacitor never reaches full charge, but after several time periods it's so close that it is 100% to >3 significant figures, I think 99.995% or 5.9997V here giving an energy of 179.98 uJ

Thanks. I didn't even come up with the idea of working out the time constant, I just thought "I don't know how to calculate voltage with respect to time" and panicked.

If you can't remember the formula, the graph is shaped like an upside down discharge graph. 5 time periods as mentioned by uber is a useful rule of thumb (didn't see he was here when I started typing