Energy stored in the capacitors

[harlock]

Is there an error in the calculation 
of the energy stored in the capacitors?

The fact is that the charge doesn't tend to zero but to the electron charge...

[John Cuthber]

In what circumstances?

[harlock]

In this istance dq >= e(electron charge) !...

[studiot]

On 11/2/2021 at 8:23 AM, harlock said:

The fact is that the charge doesn't tend to zero but to the electron charge...

 

Why should it not tend to zero ?

Zero is a valid state of charge.

Further your integral is taken from zero to Q.

 

 

[joigus]

You could call it an error. I prefer to see it as a limitation of the ideas of electrostatics when you consider charges smaller than one electron's charge, and --directly related to that-- the fact that, at some point, you need to replace classical electrodynamics with quantum electrodynamics. You cannot consider an electron as made up of elementary charges smaller than e.

Capacitance is perhaps not the best way to see it, because it's not a fundamental quantity. It doesn't make a lot of sense considering an electron as infinitely many infinitesimally-small charges adding up to give an electron. You would have, I guess, to assume what the capacity of an 'incremental electron of charge dq' is.

For a point-like electron (you haven't specified what spatial distribution of charge you're thinking of) it's best to use the expression,

\[ \frac{\varepsilon_{0}}{2}\int\left|\boldsymbol{E}\right|^{2}dV=\int_{0}^{\infty}\frac{e^{2}}{32\pi^{2}\varepsilon_{0}r^{4}}4\pi r^{2}dr \]

which gives you nonsense,

\[ U=-\frac{e^{2}}{8\pi\varepsilon_{0}}\left(\frac{1}{\infty}-\frac{1}{0}\right)=\infty \]

So, considering an electron as being made up of infinitely many (smaller) incremental charges doesn't make sense. You would like to consider a finite electron. In that case you're going to have to face even worse problems --mostly having to do with the fact that this electron cannot consistently be considered as a rigid object, nor has anybody found a way to make it elastic and be consistent with relativity --Poincaré and others tried very hard. That's why we use quantum mechanics when things get so small. And also forget about capacitance, which is a highly-derived concept.

I hope that was helpful.

2 hours ago, harlock said:

In this istance dq >= e(electron charge) !...

Sorry, I misread this.I thought you meant dq<=e, which is what you need. If you integrate from zero you should add smaller charges than e, which is what Studiot is pointing out, in a way.