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Gauss's law is one of Maxwell's four equations and provides a powerful method for calculating electric fields with symmetric charge distributions. It states: the net electric flux through any closed surface (Gaussian surface) equals the enclosed charge divided by $epsilon_0$.

$Phi_E$ = closed integral of E . dA = $Q_{enclosed}$ / $epsilon_0$

Electric flux $Phi_E$ represents the number of field lines passing through a surface. For a flat surface in uniform field: Phi = EA*cos(theta), where theta is the angle between E and the outward normal. SI unit: N $m^2$/C or V m. Dimensions: [M $L^3$ T^(-3) A^(-1)].

Critical subtleties for JEE: (1) E on the Gaussian surface is due to ALL charges (inside and outside), but the NET flux depends only on enclosed charges. External charges contribute zero net flux. (2) Zero flux does NOT mean E = 0 everywhere on the surface — it means field lines entering equal those leaving. (3) Gauss's law is always mathematically valid but only computationally useful when symmetry allows E to be factored out of the integral.

The law follows from the inverse-square nature of Coulomb's law. If the force law were 1/$r^n$ with n not equal to 2, Gauss's law would not hold.

Common flux calculations: Charge Q at center of cube — flux through each face = $\frac{Q}{6*epsilon_0}$. Charge at corner of cube — flux through that cube = $\frac{Q}{8*epsilon_0}$. Charge at center of one face — total flux through cube = $\frac{Q}{2*epsilon_0}$. These problems require careful geometric reasoning about what fraction of the total flux $\frac{Q}{epsilon_0}$ passes through the given surface.