MAXWELL'S EQUATIONS AND THE CONTINUITY EQUATION

THIS IS MAXWELL’S EQUATION These are equations which the field vectors E,D,B and H everywhere Satisfy. These equations are called Maxwell’s equation They are div D =ρf div B =0 Curl E = -dB/dt Curl H = jf + dD/dt Any possible electromagnetic field must satisfy all of Maxwell’s equation Consider the following equations Div D = ρf Curl H = jf + (𝜕D/𝜕t) These are part of Maxwell’s equation and can be combined to yield the continuity equation given below: (𝜕ρ/𝜕t) + div j =0 This is the continuity equation Consider the following identities such that for any Vector Field F div.Curl F ≡ 0 ∇.(∇^F) ≡ 0 ∇.(∇XF) ≡ 0 Applying this in the two equations above, [ the two Maxwell’s equations of interest] We have Div Curl H = div jf + div (𝜕D/𝜕t) We get this by taking the divergence of both sides of the equation So that Div Curl H = div jf + div (𝜕D/𝜕t) 0 = Div Curl H = div jf + div (𝜕D/𝜕t) {*} Differentiate the second of the Maxwell’s equation of interest and substituting in the other one (the first ie the one used earlier on;) We have: (𝜕divD/𝜕t) = (𝜕ρf/𝜕t) {**} This result in eqn {**} is then substituted in eqn {*} Ie subs for (𝜕divD/𝜕t) in * to yield 0 = div j + (𝜕ρ/𝜕t) This is the continuity equation. This shows that the continuity equation is embedded in the Maxwells equation This means that the Continuity equation can be derived from the Maxwells equation. This means that the Continuity equation is taken care of in Maxwell’s equation. HENCE ONE CAN SAY OF MAXWELL’S EQUATION THAT : Any possible electromagnetic field must satisfy all of Maxwell’s equation These are equations which the field vectors E,D,B and H everywhere Satisfy.