Faraday’s Law: The emf induced in a loop when the magnetic flux is changing in the vicinity of it

$emf= ∮\bar{E}.d \bar{l} \\ ∮E .d \bar{l} =0 \\ \text{Using Stoke’s theorem} \\ ∮\bar{E}.d \bar{l}= ∫_s^0(∇ × \bar{E} )ds \\ ∫_s^0(∇ × \bar{E} )ds=0 …..(∮E .d \bar{l}=0) \\ ∇ × \bar{E}=0$

Ampere’s Law: Magnetic field can be produced by conduction current as well as displacement current

$I= ∮ \bar{H}.d \bar{l} \\ \text{I in terms of J is given as} \\ I= ∫ \bar{J} .d \bar{s} \\ ∮ \bar{H}.d \bar{l}= ∫ \bar{J} .d \bar{s} \\ ∫(∇ × \bar{H} )ds= ∫J.ds \\ ∇ × \bar{H}=J$

Gauss Law (for Electric): The total electric flux crossing the closed surface is equal to the total charge enclosed by that surface.

$Ψ= ∫ \bar{D} .d \bar{s} \\ Q_{enclosed}= ∫ρ_v.dv \\ ∮ \bar{D} .d \bar{s}= ∫ρ_v.dv \\ ∫(∇ . \bar{D} )dv= ∫ρ_v.dv \\ ∇. \bar{D}=ρ_v \\ ∇.(ε \bar{E} )=ρ_v \\ ∇. \bar{E}=\dfrac{ρ_v}{ε} \\ ∮_s \bar{E} .ds= ∮_v \dfrac{ρ_v}{ε dv}$

Gauss’s Law (for magnetic): The total magnetic flux crossing the closed surface is zero.

$∮ \bar{B} .ds=0 \\ ∮_s \bar{B} .d \bar{s}= ∫_v(∇. \bar{B} )dθ \\ ∫_v(∇. \bar{B} )dv=0 \\ ∇. \bar{B}=0 \\ ∮_s \bar{H} .d \bar{s} =0$