Alternate derivation for energy stored by a capacitor

Question

The usual derivation of energy stored in a capacitor is as follows $$dU=Vdq\\dU=\frac QCdq$$$$U=\frac12\frac {Q^2}C\equiv\frac12QV\tag1$$ Where $V$ is the final potential. Explicitly $$V=-\int\vec E\cdot d\vec l\tag2$$ Where $\vec E$ is the net electric field (that is, this field has contributions from both plates) between the plates.

From a mathematical point of view, it makes perfect sense why a factor of $1/2$ is coming.

From a physical point of view, we can justify this factor for the parallel plate capacitor as follows.

Firstly we will distribute our system of capacitor into two sub-systems, one being the positive plate and another being the negative plate. It is important to realize that we are only interested in the interaction energy of these two systems, usually called "energy stored by the capacitor" which is given in $(1)$. Hence we will not give any regard to the way these plates were themselves created (which does not matter anyways due to the conservative nature of electrostatics) and to the energy needed to create these plates (which is infinite anyways due to infinite charge on each plate in our approximation used). Thus taking this into account we will create our capacitor as follows: Take two plates with opposite polarity very close to each other so that the net electric field is zero everywhere. Now hold the negative plate and pull the positive plate away to some distance $l$. This procedure is used specifically to simplify the calculations done below.

Consider a charge of $dq$ on the positive plate. The energy needed to bring this to $l$ is $$\Delta\phi dq\tag3$$ where $\phi$ is the potential due to the negative plate. The difference in $\phi$ is calculated at $0$ and $l$. Since we are only talking about interaction energy, the net energy needed to bring the plates close is simply $$\int\Delta\phi dq=Q\Delta\phi\tag4$$ since $\Delta\phi$ is a constant and $Q$ is $\int dq$.

Calculation of $\Delta\phi$:

The electric field due to the negative plate is $$E_n=-\frac\sigma2\tag5$$

here $\epsilon_0=1$ is used. The potential due to this, $\Delta\phi$ is $$\Delta\phi=-\int E_ndl=\frac\sigma2 l\tag6$$ Thus the energy becomes $$U=\frac\sigma2 Ql=\frac {Q^2}2\frac Al =\frac12QV\tag7$$ Where the definition of capacitance is used and $V$ is defined as in $(2)$.

How to generalize this kind of "see-through" reasoning in terms of charge distributions and electric fields as opposed to direct definitions of capacitance, to any type of capacitor?

Answer 1

Remember that charging a capacitor means shifting charge from one of its plates (initially neutral) to the other (initially neutral). Thus one plate becomes more and more positively charged while the other becomes equally and oppositely charged. The shifting is most easily done by connecting a battery across the plates. The battery pumps charge through itself from one plate to the other. The charge shifting will take a finite time because of the battery's internal resistance.

Consider the time during the charging when the capacitor has a pd $V$ between its plates and a charge $Q=CV$ (in other words charges $±Q$ on its plates). When a further charge $dQ$ passes from one plate to the other, it gains potential energy $V\ dQ$. So the extra energy gained by the capacitor is $$dU=V\ dQ=V\ CdV$$ Suppose that $V_0$ is the 'final' pd between the plates. [To all intents and purposes $V_0$ will be the battery emf.] The 'final' energy stored in the capacitor will be $$U=\int_0 ^{V_0} V\ CdV=\tfrac 12 CV_0^2$$ Intuitive justification: you are transferring a total charge $Q_0=CV_0$ from one plate to the other through a pd that was $0$ initially and $V_0$ finally, that is through a mean pd of $\tfrac 12 (V_0+0)$, so the gain in PE is $\tfrac 12 (V_0+0)\times CV_0 =\tfrac 12 CV_0^2$.