what is gauss divergence theorem?
Answers
Answer:
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem,[1] is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the tensor field inside the surface.
More precisely, the divergence theorem states that the outward flux of a tensor field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources (with sinks regarded as negative sources) gives the net flux out of a region.
The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.
In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to Green's theorem.
Answer:
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