I know the $\frac{QV}{2}$ & $QV$ stuff but I want the logic. You would say it's due to the resistance that the work gets converted to heat. But, why is it that exactly half the value of work done by the battery no matter what the $emf$ of the battery or the capacitance of the capacitor be?
Why is it so precisely half the work done by the battery?
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4Does this answer your question? What happens to half of the energy in a circuit with a capacitor? – Chemomechanics Jan 25 '21 at 04:32
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1It's "precisely half" because of the choice of using a linear model $dU=C,dV$ with constant C: Integration from 0 to V then gives a right triangle with sides V and CV that takes up half the area of a rectangular measuring V×CV, which is the energy supplied by the constant-voltage source. A more sophisticated model and a nonconstant capacitance or voltage would produce a less simple result. – Chemomechanics Jan 25 '21 at 04:43
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1Also see answers here and here on the EE site. – Chemomechanics Jan 25 '21 at 04:48
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1What does "er" mean? Is it a typo? – PM 2Ring Jan 25 '21 at 05:04
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If we have a capacitor $C$ with voltage $V$ applied across it; we will have to do $W$ work to get $Q$ inside the capacitor, the small amount of work done per small amount of charger is given by:
$$dW=VdQ$$
So the total work done on the capacitor is given by:
$$W=\int_0^Q dW$$ $$=\int_0^Q VdQ$$ $$=\int_0^Q \frac{Q}{C}dQ$$ $$=\frac{1}{2C}(Q^2-0^2)=\frac{Q^2}{2C}=\frac{QV}{2}=\frac{V^2C}{2}$$
It has nothing to do with a resistance or anything like that, it is the amount of energy stored in a capacitor and it is amount of energy taken from the battery. Remember that as a Capacitor saturates it becomes an insulator, namely $I \rightarrow 0$, so there is no more power consumed after a certain point, unlike a resistor that consumes power indefinitely.
xXx_69_SWAG_69_xXx
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so in getting the charge q on the capacitor we did qxv/2 and in bringing the charge q across the battery the battery did qxv amount of work so where is the remaining 1/2xqxv work done by the battery gone? – Rashik Jan 25 '21 at 05:31