Why would a ionised potato explode violently?

Question

I am reading Griffith's Introduction to Electrodynamics, while introducing the fundamental properties of electric charge he writes:

1. Charge comes in two varieties, which we call "plus" and "minus," because their effects tend to cancel (if you have +q and -q at the same point, electrically it is the same as having no charge there at all). [...] what if the two kinds did not tend to cancel? The extraordinary fact is that plus and minus charges occur in exactly equal amounts, to fantastic precision, in bulk matter, so that their effects are almost completely neutralized. Were it not for this, we would be subjected to enormous forces: a potato would explode violently if the cancellation were imperfect by as little as one part in $10^{10}$.

I can not figure why the potato would "explode violently". I formulated two hypotheses:

1) Local unbalance: the charge does not cancel only in the potato. Then the potato is a ionised body (hence the ionised potato in the title). I expect the potato to discharge at the first possibility, but not to blow itself apart.
Suppose - very crudely - the potato is composed only of Carbon, a modest $120\,{\rm g}$ potato contains: $$ 120\,{\rm g} = 10 \times 12\,{\rm g} = 10 \times 6.02 \times 10^{23}\,{\rm atoms} = 6.02 \times 10^{24}\,{\rm atoms}, $$ being one charge out of $10^{10}$ unbalanced, I suppose that the same fraction of the potato Carbon atoms is ionised $$ Q_{\rm potato} = \frac{6.02 \times 10^{24}}{10^{10}} \times 1.6 \times 10^{-19}\,{\rm C} = 6.38 \times 10^{-5}\,{\rm C}. $$ It does not strike me as a catastrophic amount of charge. For comparison a lightning transfers $15-350\,{\rm C}$ of electric charge [wikipedia].

2) Global unbalance: the charge does not cancel in the potato and in the environment. I imagine now that each potato has its own charge unbalance, so they start to repel / attract each other as macroscopic protons and electrons. What I can picture (in a kitchen) is an electrostatic problem with many bodies that move towards an equilibrium configuration (assuming there is no body with a disproportionate charge unbalance).

What am I not considering? What is the phenomenon that will induce a violent explosion of a potato were the electric charge not cancelled?

Answer 1

I am pretty sure that Griffiths was talking about a local imbalance.

The charge alone is not an indicator of the configuration's energy. Two point charges close enough can exceed any energy and force limit. So we need to take into account the distance between the charges and all the forces involved.

Considering a spherical potato of radius $R$ and with an uniform volume charge $\rho=\frac{3Q}{4\pi R^3}$, the electric field a distance $r<R$ from the center follows from Gauss' Law: $$\mathbf{E}=\frac{Qr}{4\pi \epsilon R^3}\hat{\mathbf{r}}.$$ Therefore, the force per unit volume is $$\mathbf{f}=\rho \mathbf{E}=\left(\frac{3Q}{4\pi R^3}\right)\left(\frac{Qr}{4\pi\epsilon R^3}\hat{\mathbf{r}}\right)=\frac{3r}{\epsilon}\left(\frac{Q}{4\pi R^3}\right)^2\hat{\mathbf{r}}.$$

Now we can compute the force exerted by the "southern" hemisphere on the "northern" one by integrating this force-per-volume over the volume of the hemisphere:

$$\newcommand\mycolv[1]{\begin{bmatrix}#1\end{bmatrix}} \int\mathbf{f}d\tau=\frac{3}{\epsilon}\left(\frac{Q}{4\pi R^3}\right)^2\int r \hat{\mathbf{r}}d\tau=\frac{3}{\epsilon}\left(\frac{Q}{4\pi R^3}\right)^2\int_0^R\int_0^{\frac{\pi}{2}}\int_0^{2\pi} r \mycolv{\sin{\theta}\cos{\phi}\\\sin{\theta}\sin{\phi}\\\cos{\theta}}r^2\sin{\theta}d\phi d\theta dr,$$ where $$d\tau=r^2\sin{\theta}drd\theta d\phi,\\\hat{\mathbf{r}}=\sin{\theta}\cos{\phi}\hat{\mathbf{i}}+\sin{\theta}\sin{\phi}\hat{\mathbf{j}}+\cos{\theta}\hat{\mathbf{k}}.$$ The force is, of course, in the direction of the $z$ axis since the $x$ and $y$ components add to zero because $\int_0^{2\pi}\cos{\phi}d\phi=\int_0^{2\pi}\sin{\phi}d\phi=0$. So we end up with: $$\mathbf{F}=\hat{\mathbf{k}}\frac{3}{\epsilon}\left(\frac{Q}{4\pi R^3}\right)^2\int_0^R\int_0^{\frac{\pi}{2}}\int_0^{2\pi} r \cos{\theta}r^2\sin{\theta}d\phi d\theta dr=\frac{1}{4\pi\epsilon}\frac{3Q^2}{16 R^2}\hat{\mathbf{k}}.$$

We are considering a 120g potato, knowing that the average density of a fresh potato is $1092 \left.\mathrm{kg}\middle/\mathrm{m}^3\right.$ its volume must be $1.1\times10^{-4} \mathrm{m}^3$. If we assume that the potato is a sphere, its radius would be $R\approx2.97\mathrm{cm}$. We also have $\epsilon=\epsilon_r\epsilon_0$, where $\epsilon_r$ is the dielectric constant of the potato. The closest thing I could find is the dielectric constant of potato starch, $\epsilon_r=1.7$. Finally, we have the total excess of charge, $Q=6.38\times 10^{-5}C$, that you nicely provided.

With all these data: $$F=\frac{1}{4\pi\cdot 1.7 \epsilon_0}\frac{3\cdot (6.38\times 10^{-5}\ \mathrm{C})^2}{16\cdot (2.97\ \mathrm{cm})^2}\approx 4.57\ \mathrm{kN}.$$

As they said in the comments, while a steel ball may support such force, a potato would basically explode.

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Now, it could be interesting to calculate the energy of the configuration: $$W=\frac{1}{2}\int Vdq=\frac{1}{2}\int \rho V d\tau=\frac{\epsilon}{2}\int_\limits{\text{all space}}E^2d\tau,$$ with $$E=\begin{cases}\frac{Qr}{4\pi\epsilon R^3}&\text{ if } r\lt R\\ \frac{Q}{4\pi\epsilon r^2}&\text{ if } r\ge R\end{cases}\ .$$ So $$W=\frac{\epsilon}{2}\frac{Q^2}{4^2\pi^2\epsilon^2}\left[\int_0^R\int_0^\pi\int_0^{2\pi}\frac{r^2}{R^6}r^2\sin(\theta)d\phi d\theta dr + \int_R^{\infty}\int_0^\pi\int_0^{2\pi}\frac{1}{r^4}r^2\sin(\theta)d\phi d\theta dr\right]=\frac{1}{4\pi\epsilon}\frac{3Q^2}{5R}\approx\frac{1}{4\pi\cdot 1.7\epsilon_0}\frac{3\cdot (6.38\times 10^{-5}\ \ \mathrm{C})^2}{5\cdot 2.97\ \mathrm{cm}}\approx 434.74\ \mathrm{J}.$$

If one of the hemispheres is held and this energy is converted completely into kinetic energy of the other, its speed would be $$v=\sqrt{\frac{2W}{m}}\approx\sqrt{\frac{2\cdot 434.74\,\mathrm{J}}{60\mathrm{g}}}\approx 120.38\ \left.\mathrm{m}\middle/\mathrm{s}\right. \approx 433.37\ \left.\mathrm{km}\middle/\mathrm{h}\right..$$ Well, I do not know much about explosions, but this looks to me like one.