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I have been having some trouble understanding the storage of energy inside a spherical capacitor composed of two concentric spherical shells where the inside shell has radius $a$ and charge $2q$, while the outside shell has radius $b$ and charge $q$. We know that

$$\Delta V=-\int\vec{E}.d\vec{r}$$

and that $U=q\Delta V$, hence we can define

$$U=-\intop_{0}^{2q}\intop_{b}^{a}\vec{E}.d\vec{r}.dq$$

Or at least I hope so. But how should we think about it? The potential difference between $a$ and $b$ is defined as the work per unit charge that an external agent has to exert in order to bring a chunk of charge, say $dq$ from $b$ to $a$. And what we are doing here I only see it as summing all of the work needed to transport the $dq$'s individually but not counting with the interaction between them.

What I'm asking is the intuition to link the previous expression and:

$$\frac{1}{4\pi\epsilon_{0}}\sum_{i=1}\sum_{j>i}\frac{q_{i}q_{j}}{r_{ij}}$$

Plus if someone could give me a hint on where this comes from I would be very appreciated

$$U=\frac{1}{2}\epsilon_{0}\int E^{2}dV$$

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  • $\begingroup$ There is an expression for the potential of an isolated sphere carrying a charge $Q$ and using that you can find the capacitance of concentric spheres carrying a charge $2q$ and an isolated sphere (the outer sphere) carrying a charge $3q$. This then enables you to find the energy stored in your arrangement of spheres. $\endgroup$ – Farcher Apr 18 '18 at 5:47

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