What is gauss's theorem ?Long answer
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Gauss’s law
Gauss’s law states that the net flux of an electric field in a closed surface is directly proportional to the enclosed electric charge.
Gauss’s law states that the net flux of a given electric field through a given closed surface, is equal to the 1/ε0 total charge enclosed by the surface.
Gauss Law Equation
Let us now study Gauss’s law through an integral equation. Gauss’s law in integral form is given below:
∫E⋅dA=Q/ε0 ….. (1)
Where,
E is the electric field vectorQ is the enclosed electric chargeε0 is the electric permittivity of free spaceA is the outward pointing normal area vector
Flux is a measure of the strength of a field passing through a surface. Electric flux is defined as
Φ=∫E⋅dA …. (2)
We can understand the electric field as flux density. Gauss’s law implies that the net electric flux through any given closed surface is zero unless the volume bounded by that surface contains a net charge.
Gauss’s law for electric fields is most easily understood by neglecting electric displacement (d). In matters, the dielectric permittivity may not be equal to the permittivity of free-space (i.e. ε≠ε0). In the matter, the density of electric charges can be separated into a “free” charge density (ρf) and a “bounded” charge density (ρb), such that:
Ρ = ρf + ρb
Gauss’s law
Gauss’s law states that the net flux of an electric field in a closed surface is directly proportional to the enclosed electric charge.
Gauss’s law states that the net flux of a given electric field through a given closed surface, is equal to the 1/ε0 total charge enclosed by the surface.
Gauss Law Equation
Let us now study Gauss’s law through an integral equation. Gauss’s law in integral form is given below:
∫E⋅dA=Q/ε0 ….. (1)
Where,
E is the electric field vectorQ is the enclosed electric chargeε0 is the electric permittivity of free spaceA is the outward pointing normal area vector
Flux is a measure of the strength of a field passing through a surface. Electric flux is defined as
Φ=∫E⋅dA …. (2)
We can understand the electric field as flux density. Gauss’s law implies that the net electric flux through any given closed surface is zero unless the volume bounded by that surface contains a net charge.
Gauss’s law for electric fields is most easily understood by neglecting electric displacement (d). In matters, the dielectric permittivity may not be equal to the permittivity of free-space (i.e. ε≠ε0). In the matter, the density of electric charges can be separated into a “free” charge density (ρf) and a “bounded” charge density (ρb), such that:
Ρ = ρf + ρb
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Explanation:
According to the Gauss law, the total flux linked with a closed surface is 1/ε0 times the charge enclosed by the closed surface.
∮E⃗ .d⃗ s=1∈0q .
According to Gauss Law,
Φ = → E.d → A
Φ = Φcurved + Φtop + Φbottom
Φ = → E . d → A = ∫E . dA cos 0 + ∫E . dA cos 90° + ∫E . dA cos 90°
Φ = ∫E . dA × 1
Due to radial symmetry, the curved surface is equidistant from the line of charge and the electric field in the surface has a constant magnitude throughout.
Φ = ∫E . dA = E ∫dA = E . 2πrl
The net charge enclosed by the surface is:
qnet = λ.l
Using Gauss theorem,
Φ = E × 2πrl = qnet/ε0 = λl/ε0
E × 2πrl = λl/ε0
E = λ/2πrε0
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