Physical Significance of Maxwell’s Equations

By means of Gauss and Stoke’s theorem we can put the field equations in integral form of hence obtain their physical significance
1.    Maxwell’s first equation is ∇. D = ρ.
Integrating this over an arbitrary volume V we get
        ∫v ∇.D dV = ∫v ρ dV.
But from Gauss Theorem, we get
        ∫s D.dS = ∫v ρ dV = q
Here, q is the net charge contained in volume V. S is the surface bounding volume V. Therefore, Maxwell’s first equation signifies that:
The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume.

2.    Maxwell’s second equations is ∇.B = 0
Integrating this over an arbitrary volume V, we get
        ∫v ∇.B = 0.

Using Gauss divergence theorem to change volume integral into surface integral, we get
        ∫s B.dS = 0.
Maxwell’s second equation signifies that:
The total outward flux of magnetic induction B through any closed surface S is equal to zero.

3.    Maxwell’s third equation is ∇ x E = - ∂B/∂t . dS
Converting the surface integral of left hand side into line integral by Stoke’s theorem, we get
        Φc E. dI = - ∫s ∂B/∂t. dS.
Maxwell’s third equation signifies that:

The electromotive force (e.m.f. e = ∫C E.dI) around a closed path is equal to negative rate of change of magnetic flux linked with the path (since magnetic flux Φ = ∫s B.dS).

4.    Maxwell’s fourth equation is
∇ x H = J + ∂D/∂t

Taking surface integral over surface S bounded by curve C, we obtain
        ∫s ∇ x H. dS = ∫s (J + ∂D/∂t) dS
 
Using Stoke’s theorem to convert surface integral on L.H.S. of above equation into line integral, we get
        Φc H.dI = ∫s (J + ∂D/∂t).dS
Maxwell’s fourth equation signifies that:

The magneto motive force (m.m.f. = Φc H. dI) around a closed path is equal to the conduction current plus displacement current through any surface bounded by the path.


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