Explain Gauss theorem.
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The Gauss Law
The Gauss law is a very convenient tool to find the electric field of a system of charges. It is especially useful when situations of symmetry can be easily exploited. The electric flux (dΦ) through a differential area is the dot product of the electric field and the area vector (this vector is normal to the area pointing towards the convex end; its magnitude is the area magnitude).
dΦ = →E . d→A = EdA cos θ
Where θ is the angle between the electric field and the area vector
The net flux through a surface is simply the integral of the differential flux dΦ,
Φ = ∫ dΦ = ∫ →E.d→A
For a closed surface such as that of a sphere, torus or a cube, this integral is represented as a loop integral
Φ = →E.d→A
The Gauss Theorem
The net flux through a closed surface is directly proportional to the net charge in the volume enclosed by the closed surface.
Φ = →E.d→A = qnet/ε0
In simple words, the Gauss law relates the ‘flow’ of electric field lines (flux) to the charges within the enclosed surface. If there are no charges enclosed by a surface, then the net electric flux remains zero. This means that the number of electric field lines entering the surface is equal to the field lines leaving the surface.
This statement also gives an important corollary: the electric flux from any closed surface is only due to the sources (positive charges) and sinks (negative charges) of electric fields enclosed by the surface. Any charges outside the surface do not contribute to the electric flux. Also, only electric charges can act as sources or sinks of electric fields. Changing magnetic fields. For example, cannot act as sources or sinks of electric fields.

The net flux for the surface on the left is non-zero as it encloses a net charge. The net flux for the surface on the right is zero since it does not enclose any charge.
The Gauss law is a very convenient tool to find the electric field of a system of charges. It is especially useful when situations of symmetry can be easily exploited. The electric flux (dΦ) through a differential area is the dot product of the electric field and the area vector (this vector is normal to the area pointing towards the convex end; its magnitude is the area magnitude).
dΦ = →E . d→A = EdA cos θ
Where θ is the angle between the electric field and the area vector
The net flux through a surface is simply the integral of the differential flux dΦ,
Φ = ∫ dΦ = ∫ →E.d→A
For a closed surface such as that of a sphere, torus or a cube, this integral is represented as a loop integral
Φ = →E.d→A
The Gauss Theorem
The net flux through a closed surface is directly proportional to the net charge in the volume enclosed by the closed surface.
Φ = →E.d→A = qnet/ε0
In simple words, the Gauss law relates the ‘flow’ of electric field lines (flux) to the charges within the enclosed surface. If there are no charges enclosed by a surface, then the net electric flux remains zero. This means that the number of electric field lines entering the surface is equal to the field lines leaving the surface.
This statement also gives an important corollary: the electric flux from any closed surface is only due to the sources (positive charges) and sinks (negative charges) of electric fields enclosed by the surface. Any charges outside the surface do not contribute to the electric flux. Also, only electric charges can act as sources or sinks of electric fields. Changing magnetic fields. For example, cannot act as sources or sinks of electric fields.

The net flux for the surface on the left is non-zero as it encloses a net charge. The net flux for the surface on the right is zero since it does not enclose any charge.
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