Chapter 1

Electrostatics and Magnetostatics

Yukawa Potential

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.

Let us consider a charge distribution with a spherical symmetry around a stationary point O.

At a certain distance , the potential is expressed by (Yukawa potential) :

Where and are positive constants.


Calculate the field at a given distance from O.

Study the two particular cases and .

What is the significance of  ?


Use the relation between the field and the potential :


The electric field is expressed by :

By noticing that, for :

It is the electric field created by a punctual charge placed in point O. This charge is the proton charge.

And, for  :

From “far away”, the field is equal to zero and the charge distribution is globally neutral.

Here, represents the order of magnitude of the dimension of the atom. It is Bohr's radius.


Calculate , the surface integral of the field exiting a sphere of radius .

Deduce that the charge distribution can be equivalent to a punctual charge placed in O and a charge located in space and defined by its volumic charge distribution .

Find the density of charge that is equivalent to the distribution.


Calculate the flux and use Gauss' law.


In spherical symmetry, the surface integral of the electric field (the flux) is given by :

Hence :

Let's consider the volume between two spheres of radius and .

Gauss' law applied to this volume gives :

We can notice :

Thus :

Hence the density of charges, supposed to represent the electron in a semi-quantic manner, is expressed by :

Thus :


The charge located in the volume between two spheres of radius and is :

We can define a linear density of charges, which is equivalent to the density of probability in quantum mechanics :

Let's compute its derivative :

We notice that when . We can verify that it is a maximum.

We see that the electron is essentially distributed around Bohr's radius.

This radius corresponds to the classical circular trajectory in the planetary model of the atom.