Question

applications of gauss theorem...

Answer 1

Applications of Gauss’s Law

Gauss’s law is useful incalculating the electric field in problems, in which it is possible to choose a closed surface such that the electric field has a normal component which is either zero or has a single fixed (constant) value at every point on the surface. Such a closed surface, for a given charge distribution, is referred to as the ‘Gaussian Surface’ for that charge distribution. We can normally find such a Gaussian surface only for charge distributions that have a high degree of symmetry associated with them. It is only when we are able to think of a suitable Gaussian surface (for a given charge distribution) that we are able to use Gauss’s theorem for a simple calculation of electric fields.
Electric Field due to a point charge (Coulomb’s Law)

We have considered Coulomb’s law as fundamental equation of electrostatics and have derived Gauss’s law from it. However Coulomb’s law can also be derived from Gauss’s law. This is done by using this law to obtain the expression for the electric field due to a point charge Consider the electric field due to a single positive point charge q. By symmetry, the field is everywhere  radial and its magnitude is the same at all points, that are at the same distance r from the charge. Hence, if we select, as a Gaussian surface, a spherical surface of radius r,  at all points on this surface and the field is radial. If we consider a small elementary area of this gaussian surface, the area vector is in the radial direction i.e. perpendicular to surface.

Gauss’s law and inverse square laws.
Gauss’s law is a direct consequence of the inverse square nature of Coulomb’s law. The two laws can be viewed as two different ways of expressing the same basic fact-the inverse square nature of the force between two point charges. This also implies that we can think of a ‘Gauss’s law’ for any force field (like the Gravitational force) that has an ‘inverse square’ nature.

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Answer 2

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