What is the Differential Form of Gauss’s Theorem

Gauss’ Law in Integral Form | Physics with Professor Matt Anderson | M18-02
Gauss’ Law in Integral Form | Physics with Professor Matt Anderson | M18-02

Gauss law is one of the fundamentals of physics. It is regarding the production of electric fields by the distribution of charges over the region.

Gauss’s law states that the flux coming out of the surface equals 1 /ϵ0 of the charge enclosed by the surface. Gauss theorem has various applications.

In this study material notes, we learn about the differential form of Gauss’s theorem.

Conditions for Gauss’s law

  • The symmetricity over charge distribution is required.
  • The electric field must be symmetric, equal, and constant at all points of the Gaussian surface.
  • The θ angle between vector A and vector E must be the same at all points of the surface.
  • The Gaussian surface must not pass through any point charge (charge must be enclosed in the Gaussian surface).

Gauss’s law equation

Integral equation of Gauss’s law:

∫E⋅dA = Q/ε0

Where,

E is the electric field vector

Q is the enclosed electric charge

ε0 is the electric permittivity of free space

A is the outward pointing normal area vector

Electric flux

Flux is a measure of the strength of a field passing through a surface. Electric flux is expressed as:

Φ = ∫E⋅dA

Flux is a scalar quantity.

Electric flux is defined as the measure of the number of electric field lines crossing an area.

The SI unit of electric flux is Nm2/C.

ΦE = E⋅S = EScosθ

Another statement of Gauss’s law states that the net flux of an electric field of a given surface divided enclosed charge should be equal to a constant.

Differential form of Gauss’s law

According to Gauss’s theorem, electric flux in a closed surface is equal to 1/ϵ0 times of charge enclosed in the surface.

Gauss law is represented by-.

Where ‘q’ represents the total charge of the surface, and ‘∈0’ represents the permittivity of free space.

Gauss theorem has different applications.

The equation gives the electric field intensity of an infinitely long charged wire

E = λ/2π∈0r

where,

E = Electric field

λ = Linear charge density

∈0 = Permittivity of free space

Where λ = q/l (where λ denotes Linear charge density, q denotes the total charge of the selected piece, and L denotes the total length of the selected piece).

The equation gives the electric field intensity due to an infinite plane sheet of charge:

E = σ/2∈0

Where,

E = Electric Field

σ = Surface charge density

∈0 = Permitivity in free space.

Where, σ = q/A (where, σ denotes surface charge density, q denotes total charge of sheet, A denotes area of sheet).

According to the differential form of Gauss’s law, the divergence of the electric field at any point in space is equal to 1/∈0 times the volume charge density ‘ρ’ at that point.

Gauss divergence theorem is represented by

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Application of Gauss’s Law

  • Here, λ linear charge density in case of an infinite line of charge, at a distance ‘d’. E = (1/4πdε0) (2π/d) = λ/2πdε0.
  • Here, σ is the surface charge density, and then the intensity of the electric field near a plane sheet of charge is represented by E = σ/2ε0K.
  • Here, the dielectric constant is K, and the medium is an AIR, then the electric field in the air = σ/Kε0. The intensity of the electric field near a plane charge conductor E = σ/Kε0 (in the medium of dielectric constant).
  • E = σ/ε0 is applied when the field is between two parallel plates of a condenser, where σ is the surface charge density.

Conclusion

Gauss law derives the relationship between the charge enclosed and the electric flux. Like Newton’s law, it is also a universal law. But it is useful under certain conditions that it is applicable to closed surfaces, the standard shape of Gaussian surfaces (spherical, cylindrical, or plane symmetry). The change distribution should be symmetric. It is not applicable for open surfaces. Gauss law is also applied to non-uniform surfaces.

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