Question

application of gauss theoram with derivation

Answer 1

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 = 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.
ϕϕ = ∮x=sx=0∮x=0x=s E.dA. ϕϕ = ∮v=Vv=0∮v=0v=V ρϵ0ρϵ0 dv ϕϕ = 1ϵ01ϵ0 ∮v=Vv=0∮v=0v=V ρρ dvSolving the above equation we have,
ϕϕ = Qϵ0Qϵ0 This 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 = QAϵ0QAϵ0 For the spherical surface A = 4 ππ r2 and
Hence the Electric field through it is,
E = Q4πr2ϵ0Q4πr2ϵ0

Gauss's Law for Magnetism
It is also one of the four Maxwell equation.
The Gauss Law for Magnetism states that there are no magnetic charges analogous to the electric charges.
The net magnetic flux out of any closed surface is zero.

Gaussian Integral Form:
ϕϕ = ∮B⃗ .dA→∮B→.dA→ = 0where, B = applied magnetic field
dA = small area Considered

Differential Form:
▽▽ . B = 0 The magnetic field is best represented by the magnetic dipole.

Although the magnetic dipole is similar to the positive and negative electric charges but magnetic monopole doesn’t exist in reality, this means that the magnetic poles exist only in dipole.