Gauss' theorem in electrostatic field

The Gauss theorem theorem in the electrostatic field states that the amount of electric flux passing through a closed surface is proportional to the amount of charge enclosed by the closed surface:
Gauss' theorem in electrostatic field

In other words, the area of ​​an electric field on a closed surface is proportional to the amount of electric charge surrounded by the closed surface.

(When the charge in the volume in question is continuously distributed, the summation strain at the right end of the equation is the integral.)

It states that the flux of the electric field intensity to any closed surface depends only on the algebraic sum of the charges in the closed surface, regardless of the position distribution of the charge within the surface, and is independent of the charge outside the closed surface. In the case of vacuum, Σq is the algebraic sum of the free charges enclosed in a closed surface. When a medium is present, Σq should be understood as the sum of free charge and polarized charge enclosed in a closed curved surface.

Gauss' Theorem reflects the fact that the electrostatic field is an active field.

Gauss' theorem is directly derived from Coulomb's law, and it completely relies on the inverse square law of the force between charges. Applying Gauss theorem to the metal conductor under the condition of electrostatic equilibrium, the conclusion that there is no net charge inside the conductor is obtained. Therefore, it is an important method to test the Coulomb's law to determine whether there is net charge inside the conductor.

When there is a dielectric in space, the above equation can also be written as
Gauss' theorem in electrostatic field

This is the total amount of free charge in the surface.

It shows that the flux of the electric displacement to any closed surface depends only on the algebraic sum of the free charge in the surface, which is independent of the distribution of the free charge and has nothing to do with the polarization charge. The electric displacement of any area of ​​electric displacement is the electric flux, and the electric displacement is also called the electric flux density. For an isotropic linear dielectric, if the entire closed surface S is in a uniform linear medium with a relative permittivity, the electrical displacement is proportional to the strength of the electric field, which is called the relative dielectric constant of the medium. A dimensionless quantity. More often encountered is the inverse problem. The charge distribution in a given area is the amount of electric field at a certain location. This problem is more difficult to resolve. Although the electrical flux through a certain closed surface is known, this information is not sufficient to determine the distribution of the electric field at each point on the surface. The electric field at any location on the closed surface may be complex. Only in the case of strong symmetry of the system, such as the electric field of uniformly charged spheres, the electric field of an infinitely uniform charged surface, and the electric field of infinitely uniform charged cylinders, the Gauss theorem in the use of electrostatic fields is more important than the principle of superposition. Simple.

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