Easiest derivation Gauss Theorem with perfect explanation | EduSpark

 GAUSS 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 is₀ 1/𝜺ₒ times the net charge enclosed by the closed surface.
         Mathematically, it can be expressed as -:
                     
                                  𝟇🇪 = ∮ₛ E. dS = q/𝜺ₒ

Proof. For the sake of simplicity, we prove Gauss's theorem for an isolated positive point charge q. Suppose the surface S is a sphere of radius r centered on q. Then surface S is a Gaussian Surface.

Q. What is Gaussian Surface ?

Ans. Any hypothetical closed surface enclosing a charge is called a Gaussian Surface of that charge.

Gauss Surface

Electric field at any point on S is

E = 1/4π𝜺₀. q/r²

This field points radially outward at all points on S. Also, any area element points radially outwards, so it is parallel to E, i.e., θ = 0º.

Therefore, Flux through area dS is 

d𝟇🇪 = E. dS = EdS cos 0º = EdS

Total flux through surface S is 

                            𝟇🇪 = ∮ₛ d𝟇🇪 = ∮ₛ E ds = E ∮ₛ dS

 = E ✖️ Total area of sphere

                                  = E = 1/4π𝜺₀. q/r². 4πr² or 

                            𝟇🇪 = q/𝜺₀

                        This proves Gauss Theorem

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