What are the applications of Gausss law



The gaussian law In electrostatics, describes the flow of the electric field of a given charge through a closed surface. Since the law can be formulated in the same way for the classical theory of gravity, it describes the flow of the gravitational acceleration field through a closed surface.

It is an application of Gauss-Ostrogradski's theorem to electrostatics or electrodynamics or, correspondingly, gravity. It is also known by this name.

Like Ampère's law, the analog for magnetism, Gauss’s law is also one of Maxwell's four equations (the first) and so fundamental for classical electrodynamics. In the case of gravitation, an equation results which is equivalent to the first Maxwell equation except for a few constants.

formulation

Prepare the integral form

The theorem uses the concept of flow, which is defined for all vector fields. Imagine a body charged with the charge Q, which is surrounded by an oriented, closed surface A (oriented means that there is an excellent outer and an inner side). The surface can have any shape, it can be a ball or a somewhat dented balloon. According to the field conception, the field lines of the charge flow through this surface, which emanate from Q, just as water flows through the surface, if there were a source or a sink within the surface.

Because the universe is electrically neutral according to the current state of knowledge, all field lines that originate from one charge must end with another, unlike charge. The flow of a charge outside of A flows in on one side and out again in another place. So the total flow only depends on the included Charge Q from. The crux of the law is that it is actually equal to Q divided by some natural constant.

The surface A can be divided into very small vectors whose amount is exactly the area of ​​the piece and whose direction is exactly perpendicular to the plane (normal vector). The flow through such a piece is the component of the vector field in the direction of the piece times the area; this is exactly what is expressed by the scalar product. The total flow through A is then the surface integral of this product over the entire surface.

With this preparation the integral formulation is now accessible:

Formulation of the integral form

The integral form of the law is

with the flux Φ, the vector field E and the enclosed charge Q. is the dielectric constant of the vacuum.

Preparation of the differential form

Instead of the total macroscopic charge Q, the charge can also be determined by the Charge density Express ρ at each point, where Q is again the volume integral of ρ over the entire volume V enclosed by A.

The divergence is, so to speak, the flow through an arbitrarily small surface (for the mathematically correct explanation, refer to the article on divergence). One then obtains using the integral form

.

Formulation of the differential form

The differential form of the law is

.

application

Gaussian law offers the possibility of specifying the generated field for a given charge distribution, often in a very simple way. The Coulomb law also follows directly.

Reference to the Poisson equation

The electric field has a potential U. The following applies:

.

If one uses the differential form, one obtains the Poisson equation.

in which the Nabla operator and is the Laplace operator.

In addition, the differential calculus provides that every solution of this equation is uniquely determined. If you have found such a U, it also describes the actual field.

Derivation of the Coulomb law

As an example, the Coulomb law for point charges and spherically symmetrical charge arrangements will be derived. Let the charge Q be given as a point charge or be distributed homogeneously over a sphere. A sphere K with the radius r around this arrangement is chosen as the surface. Let ε = 1 (very good approximation for air). The field must also be spherically symmetrical (the symmetry of the problem does not reveal anything else), i.e. it is perpendicular to the sphere everywhere. You get

and with radial symmetry

.

A homogeneously charged sphere thus creates a field as if the charge were sitting in the center.

Gravity

Within the framework of Newton's theory of gravity, the principles outlined above can also be applied to the gravitational field. The gravitational acceleration of a mass M results from the law of gravitation

.

The flow through the surface of any volume is then

,

in which is the normal vector.

Thus, the gravitational acceleration field of a mass distribution can be determined with

.

In differential form and for general mass distributions results

,

which is the gravitational equivalent of the first Maxwell equation.

Category: Electrostatics