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Electric Charges and Fields

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Gauss

According to Gauss’s law, the total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.

The total electric flux through a closed surface is zero if no charge is enclosed by the surface.

Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law. Any violation of Gauss’s law will indicate a departure from the inverse square law.

Flux through area element ΔS is,

straight ϕ space equals space straight E. increment straight S space equals space fraction numerator straight q over denominator 4 πε subscript 0 straight r squared end fraction straight r with hat on top. increment straight S
since space straight r with hat on top space and space increment straight S space have space same space direction comma
increment straight ϕ space equals space fraction numerator straight q over denominator 4 πε subscript 0 straight r squared end fraction space increment straight S
For space all space area space element comma space the space flux space will space be
straight ϕ space equals space begin inline style sum for all of end style increment straight S space fraction numerator straight q over denominator 4 πε subscript 0 straight r squared end fraction increment straight S
For space same space straight r space of space each space area space element comma
straight ϕ space equals fraction numerator straight q over denominator 4 πε subscript 0 straight r squared end fraction begin inline style sum for all of end style begin inline style increment end style begin inline style straight S end style begin inline style space end style begin inline style fraction numerator straight q over denominator 4 πε subscript 0 straight r squared end fraction end style begin inline style straight S end style begin inline style end style begin inline style Area end style begin inline style space end style begin inline style of end style begin inline style space end style begin inline style sphere end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style 4 end style begin inline style begin inline style πr end style squared end style begin inline style comma end style begin inline style end style begin inline style straight ϕ end style begin inline style space end style begin inline style equals end style begin inline style space end style straight q over straight epsilon subscript 0

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