The flux of the electric field strength passing through a closed surface is proportional to the total electric charge contained within this surface.
In science, it often happens that the same law can be formulated in different ways. By and large, nothing changes from the formulation of the law in terms of its action, however, the new formulation helps theorists interpret the law somewhat differently and test it in relation to new natural phenomena. This is exactly the case we observe with the Gauss theorem, which, in essence, is a generalization of Coulomb’s law, which, in turn, was a generalization of everything that scientists knew about electrostatic charges at the time when it was formulated.
Generally speaking, there are few areas in mathematics, physics and astronomy that were not helped by the remarkable genius of Karl Friedrich Gauss. In 1831, together with his young colleague Wilhelm Weber (18041891), he studied electricity and magnetism and soon formulated and proved a theorem named after him. To understand what its meaning is, imagine an isolated point electric charge q… Now imagine that it is surrounded by a closed surface. The shape of the surface in the theorem is not important – it can even be a deflated balloon. At each point of the surface surrounding the charge, however, an electric field formed by the charge is observed, and the product of the strength of this electric field by an arbitrarily small unit area of the surface surrounding the charge, through which the field lines of force pass, is called the flow of the electric field strength, and you can calculate the flow of tension for each surface element. Gauss’s theorem just states that the total flux of the electric field strength passing through the surface surrounding the charge is proportional to the magnitude of the charge.
The connection between Coulomb’s law and Gauss’s theorem will become apparent with a simple example. Suppose the charge q surrounded by a sphere of radius r. At a distance r from the charge, the strength of the electric field, which is determined by the force of attraction or repulsion of a unit charge placed at the corresponding point, will be, according to Coulomb’s law:
E = kq/r^{2}
And we get the same value for any point of the sphere of a given radius. Consequently, the total flux of the electric field strength will be equal to the value of the field strength at a distance r from the charge multiplied by the area of the sphere (which, as you know, is equal to 4πr^{2}). In other words, the total flow will be equal to:
4πr^{2} × kq/r^{2} = 4πkq
This is Gauss’s theorem.
An interesting consequence of it is obtained if we apply this theorem to a solid metal. Imagine a solid metal object and an imaginary closed surface within it. The total electric charge inside such a surface will be zero, since inside there will be an equal number of positive and negative charges – protons of atomic nuclei and electrons, respectively. Consequently, the flux of the electric field strength passing through such a closed surface will also be zero. Since this is true for any closed surface inside the metal, this means that inside metal does not exist and the electric field cannot exist.
This property of metals is often used by experimenters and communications engineers to protect highly sensitive instruments from externally induced electrical noise. Usually the device is simply surrounded by a protective copper shield. According to the Gauss theorem, external electric fields are simply not able to penetrate into such a shell and interfere with the operation of the device.
Another interesting consequence of the Gauss theorem is that if you are caught by a thunderstorm on the road, the safest thing for you is not to get out of the car, because there you are surrounded by an allmetal screen. Even if lightning strikes your car, nothing will threaten you inside, since the entire discharge will pass through the body and go into the ground. The rubber will most likely burn out, but you yourself will remain safe and sound.
1931

Magnetic monopoles
