State Gauss’s theorem.How Coulomb’s law can be derived from it ?

Explanation

Deduction of Coulomb.s law from Gauss.s theorem : Let a charge + q is placed at O Taking O as centre, draw a gaussian surface of radius r.
Now consider a small elementary area dS at a point P on the surface. Here, OP = r. Now flux through area dS is given by
`d phi = vec(E ). vec(dS)`
and , total flux through whole surface,
`oint d phi_(E) = oint vec(E).vec(dS) =oint vec(E). hat(n) dS`
where `hat(n)` is a unit vector normal to area dS
`therefore phi_(E) = oint E dS cos 0^(@)`
or `phi_(E) = oint E dS`
or `phi_(E) = oint E dS`
Using Gauss.s theorem,
`phi_(E) = (1)/(epsilon_(0)) q`
Comparing eqns (1) and (2), we get
`E oint dS = (q)/(epsilon_(0)) or E (4 pi r^(2)) = (q)/(epsilon_(0))`
`[therefore oint dS` = area of the spherical surface of radius `r = 4 pi r^(2)`]
or `E = (q)/(4 pi epsilon_(0)r^(2))`
Eqn. (3) gives the magnitude of electri intensity at point P due to the charge q. If we place a test charge `q_(0)` at point P, then magnitude of force on `q_(0)` is given by
`F = q_(0)E`
= `q_(0) (1)/(4 pi epsilon_(0)) (q)/(r^(2))`
or `F = (1)/(4 pi epsilon_(0)) (qq_(0))/(r^(2))`
which is Coulomb.s law,
In vector notation,
`vec(F) = (1)/(4 pi epsilon_(0)) (qq_(0))/(r^(2)) hat(r )`