Chapter 4 --- The electric field inside of the electron

Chapter
4

The electric field inside of
the electron

As we know, the Gauss Law of Electric Field defines the relationship between electric charge and electric field as follows:

(4.1)

In which is the electric field strength, and is the electric charge density.

If we combine the Gauss Law of Electric Field equation (4.1) and the electric charge density distribution equation (2.1), we will obtain the follows electric field solution:

(4.2)

Below, we will prove that the above equation (4.2) satisfies the Gauss Law of Electric Field.

As we know the gradient in spherical coordinate is:

From equation (4.2), thus:

(4.3)

Thus:

(4.4)

(4.5)

Therefore:

(4.6)

And so:

(4.7)

If we combine equation (4.7) and (2.1), then we will get the Gauss Law of Electric Field.

Thus we have proved that the electric field equation (4.2) indeed satisfies the Gauss Law of Electric Field.