(i) Let us imagine an arbitrary volume V enclosed by a closed surfaceS . Now, integrate both sides of the first equation ∆.D = pà ∆’.E = p/ε0 relative to the volume V.
Then, we get
Which shows that the total electric flux over a closed surface S is – times the total charge En enclosed by the closed surface S. This is nothing but Gauss’s law in electrostatics.
(ii) Integrate both sides of the second equation V’.B = 0 relative to an arbitrary closed volume V enclosed by a closed surface S.
Which states that the total magnetic flux over a closed surface is zero. This confirms the non-existence of magnetic monopole.
(iii) Now, consider an open surface (two-sided) S with boundary a simple closed curve c. Let us integrate both sides of third equation v XE = – aaB with respect to the I t open surface S.
The left hand side represents electromotive force and right hand side is time rate of change of magnetic flux. Thus, this equation physically signifies that the electromotive force(e.mf) induced in a closed circuit is equal to the time rate of loss of magnetic flux of the surface of the circuit. This is Faraday’s law of electromagnetic induction in association with Lenz’s law.
(i) (iv) If we, integrate both sides of the fourth equation ∆ x B =µ [J + Jd] relative to an open surface S having boundary a simple closed curve C, we get
ʃʃS (∆.B) .ds =ʃ ʃS µ (J +J d ) .ds
This shows that the line integral of magnetic induction vector over a closed circuit is equal to µ times the total current (conduction plus displacement current) included in the circuit. This is simply “modified Ampere’s circuital law”.