Hello again,
I'm very sorry as i typed the Latex version of the equation improperly. I'm correcting that first then we'll get to the final question about the charge.
The energy stored in a capacitor is:
E=(C*v^2)/2
[LATEX]E=(1/2)*C/v^2[/LATEX]
and the charge transferred between times t1 and t2 is:
q=integral(t1 to t2) i dt
[LATEX]q=\int _{t1}^{t2} {i}\ dt [/LATEX]
so the charge is basically the integral of the current over the time period the charge is being transferred. If we happen to have constant current then the charge is just the current times the time period:
q=i*t
CORRECTION:
The energy stored in a capacitor is:
E=(C*v^2)/2
[LATEX]E=(1/2)*C*v^2[/LATEX]
Luckily you didnt have a problem with that part anyway.
The charge stored is:
[LATEX]q=\int _{t1}^{t2} {i}\ dt [/LATEX]
and all that means is that we sum all the infinitesimal contributions of the current 'i' over time, and the sum is the total charge transferred. It's simpler to look at a constant current and also when the current changes in discrete steps. Lets do that.
Do you understand how to integrate?
Say we have a current that is 1 amp for 2 seconds (starting from t=0) and then the current changes to 2 amps for 3 seconds, then goes to zero. To find the total charge, we multiply the current times the time for each interval and then sum all the results.
The first interval has 1 amp for 2 seconds, so that comes out to:
q1=1*2=2
The second interval has 2 amps for 3 seconds, so that comes out to:
q2=2*3=6
Now we sum the results:
q=q1+q2=2+6=8
So the total charge transferred is 8 Coulombs.
Make sense so far?
Now lets say those intervals were shorter, 0.2 seconds and 0.3 seconds respectively.
For interval 1 we have:
q1=1*0.2=0.2
and for interval 2 we have:
q2=2*0.3=0.6
and the total now is:
q=0.2+0.6=0.8 Coulombs.
It follows that if we let the intervals get even smaller, the total would get smaller too, so for intervals of 0.002 and 0.003 seconds we would have a total of:
q=0.002+0.006=0.008 Coulombs.
It just so happens that a continuous time signal could be broken up into small time units (say 1ms each) and we could approximate by summing all the contributions over the entire wave for time increments of 1ms. We might then have a reasonable approximation. But for any wave if we let the time increments get extremely small like 1e-12 then we get a more accurate result, and if we let them get even smaller like 1e-24 then we'd get an even more accurate result than before. Ultimately when we let the intervals tend to zero we get an exact value, and that is equal to the integral above.
Does this make sense?