# What if… you had a bowl of electrons? [closed]

My chemistry teacher used to tell us that if you had a soup bowl with only electrons in it, the explosion could make you fly to Pluto. Was he right? Could this happen?

## closed as off-topic by AccidentalFourierTransform, Aaron Stevens, WillO, stafusa, Emilio PisantyOct 1 '18 at 6:20

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• A very similar question is discussed in what-if what-if.xkcd.com/140. Let's say that if the whole moon were to be turned into electrons, the consequences would be slightly more severe. – csiz Oct 1 '18 at 1:08
• @csiz I thought I had read a scenario like this but I couldn't remember where! Thanks for putting the link up. – Aaron Stevens Oct 1 '18 at 1:17

The answer would depend how densely the electrons are packed. Let's say we have 1 kg of electrons, meaning we would have about $$N = 10^{30}$$ of them. For simplicity, let's approximate by arranging all of these electrons arranged in a spherical shell of radius $$r=0.1$$ meters. By symmetry, the voltage at the location of each charge would be constant and would have the following value.

$$V = -\frac{1}{4\pi \epsilon_0}\frac{Ne}{r}$$

We can thus compute the potential energy $$U = \frac{1}{2}Q V$$ associated with this configuration.

$$U = \frac{1}{2} \cdot Ne \cdot \frac{1}{4\pi \epsilon_0}\frac{Ne}{r} = \frac{N^2 e^2}{8 \pi \epsilon_0 r} \approx \boxed{1.15 \times 10^{33} \text{ Joules}}$$

Ignoring air friction, the Earth's escape velocity is $$v =11200$$ meters / second, which for a $$m=2\times 10^6$$ kg space shuttle, would only require $$E = \frac{1}{2} mv^2 = 1.25 \times 10^{14}$$ Joules.

So, yeah, you'd have way more than plenty sufficient energy.

• good job, Trevor. simple and direct. – niels nielsen Sep 30 '18 at 22:13
• Thanks for the answer! So you'd need less than $1\mu$g of electrons to get to Pluto :) – CuriousStudent Sep 30 '18 at 22:30
• @AaronStevens No, but if you integrate over all of the charge it is as if they were. A charged spherical shell acts as if it were all concentrated at the center as far as electric fields outside of the shell are concerned. – Trevor Kafka Sep 30 '18 at 23:04
• I'm not sure there would be anything left you could call "you" that would be making it to Pluto – Nacht Sep 30 '18 at 23:37
• It's important to note that even 1kg of electrons is a lot, a 150kg human has less than about 50 grams of electrons total. – teclnol Sep 30 '18 at 23:49

A uniform sphere of radius $$r$$ and total charge $$Q$$ has an electric potential energy of $$3Q^2/20\pi \epsilon_0 r$$.

Let's say your bowl is like a sphere of radius 5 cm. If it's all water ( molecular weight 18$$m_u$$) and $$\mu= 9/5$$ of a mass unit for every electron, then the number of electrons is $$N_e = 1000 \frac{4\pi r^3}{3}/\mu m_u = 5\times 10^{26}$$

If all the nuclei were removed, the electric potential energy would be $$7\times 10^{26}$$ Joules.

Pluto is at about 40 au from the Sun, effectively to get there, you have to escape from Earth and escape from the Sun. To do this you would need to launch from the Earth with a carefully directed speed of at least 16 km/s. If your mass is 100 kg, that takes a mere $$1.3\times 10^{10}$$ J.