Gauss' law is also called Gauss' flux theory, or called divergence theorem, Gauss divergence theorem, Gauss Ostrogradsky formula, Ostrogorski theorem or Gauss Austrian formula.stayElectrostaticsGauss theorem indicates thatchargeSum and generatedelectric fieldOn the closed surfacefluxThe relationship between integrals.
Gauss' law shows the relationship between the charge distribution in the closed surface and the generated electric field.Gauss theorem inelectrostatic fieldAnalogous to those applied in magneticsAmpere's law, and both are concentrated inMaxwell's equationsMedium.Because of the mathSimilarityGauss law can also be applied to other physical quantities determined by the inverse square law, such asgravitationperhapsirradiance。
That is, the flux of the vector passing through any closed surface is equal tovectorOfdivergenceThe integral of the volume enclosed by a closed surface.It gives the integral transformation relationship between the closed surface integral and the corresponding volume integral. It is an important identity in vector analysis and one of the important formulas for studying fields.
Physical applications
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vector analysis
Gauss theorem is one of the important theorems in vector analysis.It can be expressed as:[2]
This formula is independent of the choice of coordinate system.
(When the charge in the volume is continuously distributed, the summation strain at the right end of the above equation is integral.)
It means that the flux of electric field intensity to any closed surface only depends on the algebraic sum of charges in the closed surface andchargeThe position distribution of is independent of the electric charge outside the closed surface.Under vacuum,ΣqIs enclosed in a closed surfaceFree chargeThe algebraic sum of.When media is present,ΣqIt should be understood as the sum of free charges and polarized charges enclosed in a closed surface.
Gauss theorem is derived directly from Coulomb's law, which completely depends on the inverse square law of the force between charges.Apply Gauss theorem toElectrostatic balanceThe conclusion that there is no net charge inside the conductor is obtained for the metal conductor under the condition ofCoulomb's lawImportant methods.
When there is dielectric in the space, the above formula can also be recorded as[3]
Where
Is the total amount of free charge in the surface.
It shows that the flux of electric displacement to any closed surface only depends on theAlgebraand
It has nothing to do with the distribution of free charge and polarization charge.The energy of electric displacement to any area is electric flux, so electric displacement is also called electric flux density.For isotropic linear dielectric, if the entire closed surfaceSAt a uniform relative dielectric constant of
In linear medium, the electric displacement is proportional to the electric field strength,
More often, we encounter the inverse problem.The quantity of charge distribution in a given area is the electric field at a certain position.This problem is difficult to resolve.Although the electric flux passing through a closed surface is known, this information is not enough to determine the electric field distribution at each point on the surface, and the electric field at any position on the closed surface may be very complex.Only when the system has strong symmetry, such as the electric field of a uniformly charged sphere, the electric field of an infinitely large uniformly charged surface, and the electric field of an infinitely long uniformly charged cylinder, can the use of the Gauss theorem be better than the use of thesuperposition principle Easier[4]。
magnetic field
Gauss theorem of magnetic field points out that no matter for stable magnetic field or time-varying magnetic field, there are always:[3]
becauseMagnetic line of forceThe curve is always closed, so any magnetic line of force entering a closed surface must come out from the inside of the surface, otherwise the magnetic line of force will not be closed.If for a closed surface, the outward direction is defined as the direction of the positive normalMagnetic fluxIs negative, the magnetic flux is positive, then the total magnetic flux passing through a closed surface is 0.This law is similar to the Gauss theorem in electric field, so it is also called the Gauss theorem.
Electrostatic field and magnetic field
There are essential differences between the two.stayelectrostatic fieldIn, because there are independent charges in nature, the electric field line has a starting point and an end point. As long as there is a net positive (or negative) charge in the closed surface, the electric flux passing through the closed surface is not equal to zero, that is, the electrostatic field is an active field;In the magnetic field, there is noMagnetic monopoleYes, N-pole and S-pole cannot be separated,Magnetic induction lineThey are closed lines without head and tail, so the magnetic flux passing through any closed surface must be equal to zero.
In mathematics
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Gauss Theorem 2
(Basic Theorem of Algebra)
Theorem: All rational integral equations
There is at least one root.
Inference: unary n-degree equation
There are only n roots (includingimaginary rootAnd multiple roots).
Gauss Theorem 3
(Number Theory)
A necessary and sufficient condition for positive integer n to be expressed as the sum of squares of two integers[1]All forms of n are like 4k+3shapeThe power of prime factors of is even.
