State gauss's theorem derive coulomb's inverse …

Gauss's Theorem: According to Gauss's theorem total number of electric lines of force passing normally through a closed surface of ray shape in an electric field(i.e., total electric flux) is equal to 1/εo​​ times the total charge present within that surface. i.e., ΦE​=εo​q​ where, εo​= permitivity if free space, q in vaccum. Derivation of Gauss's Theorem: Let +q charge is placed at a point O and a point P lies at distance r from the point O. Imagine a sphere of radius r and centre O. Thus, point P lies on the surface of the sphere. Now, the surface of the sphere will be have as Gaussian surface. Therefore, the intensity of electric field on the surface at all the points will be equal in magnitude and will be directed radially outward. ∴ The electric flux passing through the spherical surface. Φ=E.S.cos0o Where S is surface area i.e., S=4πr2= Surface area of sphere ∴ΦE​=E.S.         or ΦE​=4πr2E          .........(i) But, by Gauss, theorem, we have ΦE​=εo​q​      .....(ii) Hence, from equation (i) and (ii), we get E4πr2=εo​q​ or E=4πεo​r2q​ Now, imagines a charge qo​ placed at point P. ∴Force on qo​, F=qo​E F=4πεo​r2qo​q​ F=4πεo​1​⋅r2qo​q​. Which is Coulomb's inverse square law.