Gauss's Theorem:
This theorem gives a relationship between the total flux passing through any closed surface and the net charge enclosed within the surface.
Gauss theorem states that the total flux through a closed surface is 1/ϵo times the net charge enclosed by the closed surface.
Mathematically.it can be expressed as
ϕE=∮.SE⃗ .dS−→=qϵ0
Proof:
We prove Gauss's theorem for an isolated positive point charge q. As shown in fig suppose the surface S is a sphere of radius r centered on q. Then surface S is a Gaussian Surface.
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| Flux through a sphere enclosing a point charge |
The electric field at any point on S is
E=14πϵ0.qr2
This field points radially outwards at all points on S. Also, any area elements point radially outwards, so it is parallel to Ē i.e. θ=0.
∴ Flux through area dS −→
dϕE=E⃗ .dS−→=EdScos00=EdS
Total flux through surface S is
ϕE=∮.SdEE=∮.SEdS=E∮.SdS
= E x ( Total area of the sphere)
=14πϵ0.qr2.4πr2
ϕE=qϵ0
Knowledge Point:
- Gauss's theorem is valid for a closed surface of any shape for any general charge distribution.
- The charge q appearing in the Gauss's theorem includes the sum of all the charges located anywhere inside the closed surface.
- The electric field E appearing in Gauss's theorem is due to all the charges, both inside and outside the closed surfaces. However, the charge q appearing in the theorem is only contained within the closed surface.
- Gauss's theorem is based on the inverse square dependence on distance contained in the coulomb's law. In fact, it is applicable to any hold in case of any departure from inverse square law.
- Any hypothetical closed surface enclosing a charge is called the Gaussian surface of that charge.
Very good notes
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