State gauss theorem in electrostatics and express it mathematically
Answers
Answered by
8
We know that the electric field of surface is given by Coulomb’s law. But for closed surface, the distribution of electric charge which produces the electricity is given by Gauss law. This is also known as flux theorem. This is one of the important laws which are helpful for determining amount of charge that present on closed surface and for calculation of electric field. The basis of classical electrodynamics is given by Maxwell and Gauss law is one of them. The other three equations of Maxwell are the induction law of Faraday, law of ampere with modification of Maxwell, and the magnetism law of Gauss.
Gauss law is also useful for derivation of Coulomb’s law of electric field. The mathematical form of this law is given by vector calculus. It can be in integral and differential form which is related the divergence theorem. Here, we are discussing about both the law of Gauss related to electricity and magnetism and its mathematical derivation with its examples. Let’s discuss complete description of this.Gauss's law states that the enclosed electric charge in a closed surface is proportional to the net flux of an electric field through the surface.
The Gauss's law is one of the Maxwell law of electromagnetism and it relates to the electric fields at points on the Gaussian (closed) surface and the net charge enclosed by that surface.
The flux of the electric field passing through a closed surface is defined as the product of the electric field passing through the area and the area of the surface in a plane perpendicular to the field.
In other words, Gauss law is also defined as the total charge Q enclosed within a surface divided by dielectric constant.
Hence, Gauss law can be mathematically written as,
ϕϕ = Qϵ0Qϵ0
Where,
ϕϕ = Electric flux through a given surface,
Q = total charge within a given surface,
ϵ0ϵ0 = Electric constant.
From the equation (1) we can also state that the electric flux flowing outwards to the surface is proportional to the total electric charge enclosed by it.
The Gauss law can also be mathematically defined as surface integral of the Electric field through the given surface,
ϕϕ = ∮x=sx=0∮x=0x=s E.dA................(2)
where,
E = electric field,
dA = infinitesimal element of area of the enclosed surface.
The above equation (2) can also be written by the divergence theorem as
▽▽ . E = ρϵ0ρϵ0 ....................(3)
where,
▽▽. E = electric field divergence,
ρρ = total electric charge density.
Derivation of Gauss's Law
Gauss law is also defined as the total charge Q enclosed within a surface divided by dielectric constant.
Hence Gauss law can be mathematically written as,
ϕϕ = Qϵ0Qϵ0Where,
ϕϕ = Electric flux through a given surface,
Q = total charge within a given surface,
ϵ0ϵ0 = Electric constant.
From the equation (1) we can also state that the electric flux flowing outwards to the surface is proportional to the total electric charge enclosed by it.
The Gauss law can also be mathematically defined as surface integral of the electric field through the given surface,
ϕϕ = ∮x=sx=0∮x=0x=s E.dA ................(2)where,
E = electric field,
dA = infinitesimal element of area of the enclosed surface.
The above equation (2) can also be written by the divergence theorem as,
▽▽ .E = ρϵ0ρϵ0 ..............(3)where,
▽▽ .E = electric field divergence,
ρρ = Total electric charge density.
ϕϕ = ∮x=sx=0∮x=0x=s E.dA.ϕϕ = ∮v=Vv=0∮v=0v=V ρϵ0ρϵ0 dvϕϕ = 1ϵ01ϵ0 ∮v=Vv=0∮v=0v=V ρρdvSolving the above equation we have,
ϕϕ = Qϵ0Qϵ0This is the same equation we have already discussed and represented in equation (1).
The electric field passing through the given surface is given by
ϕϕ = Qϵ0Qϵ0
= E. A
E = QAϵ0QAϵ0For the spherical surface A = 4 ππ r2 and
Hence the Electric field through it is,
E = Q4πr2ϵ0Q4πr2ϵ0
Gauss law is also useful for derivation of Coulomb’s law of electric field. The mathematical form of this law is given by vector calculus. It can be in integral and differential form which is related the divergence theorem. Here, we are discussing about both the law of Gauss related to electricity and magnetism and its mathematical derivation with its examples. Let’s discuss complete description of this.Gauss's law states that the enclosed electric charge in a closed surface is proportional to the net flux of an electric field through the surface.
The Gauss's law is one of the Maxwell law of electromagnetism and it relates to the electric fields at points on the Gaussian (closed) surface and the net charge enclosed by that surface.
The flux of the electric field passing through a closed surface is defined as the product of the electric field passing through the area and the area of the surface in a plane perpendicular to the field.
In other words, Gauss law is also defined as the total charge Q enclosed within a surface divided by dielectric constant.
Hence, Gauss law can be mathematically written as,
ϕϕ = Qϵ0Qϵ0
Where,
ϕϕ = Electric flux through a given surface,
Q = total charge within a given surface,
ϵ0ϵ0 = Electric constant.
From the equation (1) we can also state that the electric flux flowing outwards to the surface is proportional to the total electric charge enclosed by it.
The Gauss law can also be mathematically defined as surface integral of the Electric field through the given surface,
ϕϕ = ∮x=sx=0∮x=0x=s E.dA................(2)
where,
E = electric field,
dA = infinitesimal element of area of the enclosed surface.
The above equation (2) can also be written by the divergence theorem as
▽▽ . E = ρϵ0ρϵ0 ....................(3)
where,
▽▽. E = electric field divergence,
ρρ = total electric charge density.
Derivation of Gauss's Law
Gauss law is also defined as the total charge Q enclosed within a surface divided by dielectric constant.
Hence Gauss law can be mathematically written as,
ϕϕ = Qϵ0Qϵ0Where,
ϕϕ = Electric flux through a given surface,
Q = total charge within a given surface,
ϵ0ϵ0 = Electric constant.
From the equation (1) we can also state that the electric flux flowing outwards to the surface is proportional to the total electric charge enclosed by it.
The Gauss law can also be mathematically defined as surface integral of the electric field through the given surface,
ϕϕ = ∮x=sx=0∮x=0x=s E.dA ................(2)where,
E = electric field,
dA = infinitesimal element of area of the enclosed surface.
The above equation (2) can also be written by the divergence theorem as,
▽▽ .E = ρϵ0ρϵ0 ..............(3)where,
▽▽ .E = electric field divergence,
ρρ = Total electric charge density.
ϕϕ = ∮x=sx=0∮x=0x=s E.dA.ϕϕ = ∮v=Vv=0∮v=0v=V ρϵ0ρϵ0 dvϕϕ = 1ϵ01ϵ0 ∮v=Vv=0∮v=0v=V ρρdvSolving the above equation we have,
ϕϕ = Qϵ0Qϵ0This is the same equation we have already discussed and represented in equation (1).
The electric field passing through the given surface is given by
ϕϕ = Qϵ0Qϵ0
= E. A
E = QAϵ0QAϵ0For the spherical surface A = 4 ππ r2 and
Hence the Electric field through it is,
E = Q4πr2ϵ0Q4πr2ϵ0
Answered by
6
Answer; it state thae the net flux of an electric field in a closed surface is directly proportional to the enclose electric charge.
Step-by-step explanation:
mathematica Q= ΦE•
therefore gauss's law formula express as
Φ=Q/E•
where Q = total charge within the given surface.
E•= electric constant
Similar questions