Hey i have a new proof of gauss law.now i am listing it below
let there be an arbitaraly shaped body.let there be a charge Q,e='epsilon not',E=electric field,dS=small surface element.
let the body be consisting of many small surface elements dS_{1},dS_{2}......
let the individual surfaces be so small that radius here are constant and be r_{1},r_{2}......... respectively
therefore surface integral of E.dS=E_{1}.dS_{1}+E_{2}.dS_{2}+.....
since theta =0 therefore E.dS=E*dS
therefore putting values
Q*dS_{1}/(4*pi*e*r_{1}^{2})+Q*dS_{2}/(4*pi*e*r_{2}^{2})+....
d(theta_{1})=dS_{1}/dr_{1}^{2},d(theta_{2})=dS_{2}/dr_{2}^{2},etc (from definition of solid angle)
taking everything in common
net elctric flux=Q/(4*pi*e)(dS_{1}/dr_{1}^{2}+dS_{2}/dr_{2}^{2}+...)
=Q/(4*pi*e)(d(theta_{1})+d(theta_{2})+....)
=Q/(4*pi*e)(4*pi)=Q/(e) (since sum of all solid angle around the charge is 4*pi)
