State and prove Gauss's Theorem.
Gauss's Theorem: The net electric flux passing through any closed surface is 1εo times, the total charge q present inside it.
Mathematically, Φ=1εo⋅q
Proof: Let a charge q be situated at a point O within a closed surface S as shown. Point P is situated on the closed surface at a distance r from O. The intensity of electric field at point P will be
¯E=14πεo⋅qr2 .......(1)
Electric flux passing through area ds enclosing point P,
dΦ=→E⋅→ds
or dΦ=E⋅dscosθ
[where θ is the angle between E and ds]
Flux passing through the whole surface S,
∫∫sdΦ=∫∫sE⋅dscosθ ........(2)
Substituting the value of E from eqn. (1) in eqn. (2),Φ=∫∫s14πεoqr2dscosθ [Hereint∫sdΦ=Φ]Φ=14πεoq∫∫sdscosθr2⇒Φ=14πεoq⋅ω [∵∫∫sdscosθr2=ω]Here ω= solid angle.
But here the solid angle subtended by the closed surface S at O is 4π, thus
Φ=14πεo×q×4π
or Φ=1εo⋅q.