Does the energy inside a capacitor has a variable dependency of the capacity?

Question

While studying energy inside capacitors, I came up with something: If we can calculate the energy inside it as:

$$U=\frac{1}{2}QV=\frac{1}{2}CV^2=\frac{1}{2}\frac{Q^2}{C}$$

Then, why in some cases it appears that the energy depends directly or inversely proportional to the capacity mathematically (second and third form of the equation)

I know it have to do with the dependency of the energy about the charge Q, the potential V and the capacity C, but how could I understand it in a formal way?

I took the freedom to correct your last term, it had the wrong exponents. Insert $Q=CV$ in the equations above and you will see that you are getting equivalences. In other words: for a given capacitance you can not chose $Q$ and $V$ independently. If you want to keep the same charge on a smaller capacitor, then you need more voltage. — CuriousOne, Apr 29 '16 at 03:34
Thank you correcting my slip. I already understand that C does not depends of V and Q, as it is a constant that only depends of the properties of the material. My question was asking more about why mathematically it appears that the energy initially depends linearly of the capacity, and at the same time, it could be shown like it depends inversely to it. — Luis_V, Apr 29 '16 at 03:57
$C$ does not depend on the other two quantities, but it still links them, so you can't treat them independently. — CuriousOne, Apr 29 '16 at 04:04

score 4 · Accepted Answer · answered Apr 29 '16 at 05:56

$C$ is a property of the system. It connects the Voltage across the capacitor $V$ and the charge $Q$ stored in it. You can do the same thing with a spring using the system equivalence between a capacitor and a spring. The conclusion to draw from your relations is this.

1.) Suppose you have a lot of capacitors with different $C$ and you apply the same potential across all of them. Then if you measure the energy stored in these capacitors you will find that the energy increases linearly with the capacitance.

2.) Now suppose you take these capacitors and adjust the potential across each of them so that all of them end up with the same amount of charge $Q$. Then if you measure the energy stored in them you will find that the energy varies inversely with $C$

In short the concept of proportionality holds only when the other parameters in the equation doesn't vary. So one relation holds when $Q$ is constant and the other holds when $V$ is constant.

Just the answer I was looking for. Thank you very much – Luis_V Apr 30 '16 at 16:14 — Luis_V, Apr 30 '16 at 16:14

score 0 · Answer 2 · edited Jun 11 '20 at 09:33

0

The 2nd and 3rd forms of the equation comes from the following relation

$$\boxed{\boldsymbol {C= \frac {Q}{V}}} $$

Where $\boldsymbol {C}$ is the capacitance, $\boldsymbol {Q}$ is the charge and $\boldsymbol {V}$ is the potential.

The formula for Energy stored in the the capacitor is

$$\boxed{U=\frac{1}{2}QV} $$

Replacing $Q$ with $CV$, from the first relation we get,

$\frac{1}{2}CV²$

Now replacing $V$ with $Q/C$, from the first relation we have,

$\frac{1}{2}Q^2/C$