Chapter 1

# Electrostatic energy of an electron

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

An electron is considered as a sphere of radius in which the charge is evenly distributed, with the density .

We additionally suppose all the energy is in an electrostatic form.

We give :

and (electron's mass).

## Question

Compute the electrostatic energy of the electron.

Deduce an order of magnitude of .

### Hint

Imagine the electron is like an onion. It was built by bringing thin spherical and concentric layers from the infinity ...

### Solution

Let's establish the expression of the electrostatic energy of a sphere of radius .

It is evenly charged in volume, has total charge , and a density of charges .

Let's build this sphere in a reversible manner. We bring the charge , from the infinity.

Its potential increases from zero to the potential of the sphere, under construction, of radius , given by :

The elementary work required is :

Hence :

To get an order of magnitude of , we must identify the electrostatic energy calculated to the energy of mass given by Einstein's relativity :

So :

#### Remarque :

By analogy, the gravitational energy of a star (or planet) of mass and radius can be deduced :

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