all in all: My understanding of Maxwell's Equations

Every Professor i asked, they said i should have basic understanding of the Maxwell's equation to be able to understand the propagation of waves. So I went out searching the meaning of the well known Maxwell's equation. Sure it is easy to just write the mathematical interpretation but this time i really wanted to know their physical implication as well.

So what are Maxwell's equations, to answer this we have to be able to know certain laws. Not in any order, the first is Ampere's law. Basically what i understood is that current carrying conductor will have magnetic field around it. for a simple conductor carrying current, the direction of the magnetic field is obtained by right hand grip with thumb pointing in the direction of the current. Now conversely if we integrate the Magnetic field around the close loop we get Current.

$\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_S(\mathbf D)}{\partial t}$

now the second term out here we can see was added by Maxwell. This term accounts the effect of time changing displacement current.

The second law is Faraday's. Which was little easy to understand. We know that electric potential between two point is the amount of work done in moving a unit positive charge from one point to the other against the electric field. Now If we try to find the work done in moving a charge in a loop then the total work will be zero since we end up getting at the very spot we started from. So line integral of electric field around a close loop will be equal to zero. But this is the case only for time invariant field. For time variant magnetic fields the line integral of electric field will yield the EMF. And if we take the curl of the electric field, which is basically the measure of varying electric field, gives us rate of change of magnetic flux with time.

$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_S{(\mathbf B)}}{\partial t}$

the third is Gauss law. Now if D is electric flux density then it's surface integral will give us the charge enclosed itself.

$\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V)$

And the fourth law is of Gauss too but for magnetic fields. Since magnetic field always follow a close loop. and there hasn't been any discovery of magnetic mono-pole the interpretation of this equation was self evident.

$\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0$

So far i've gained little idea about the equations and am still striving to have a deeper understanding regarding the implications. The equations written above are in integral form. The differential form of the Maxwell equations were yet simpler to analyze apart from few, by few i meant all four of them.

The differential form of ampere was that curl of magnetic field gave current density and time-changing displacement current. I've been trying to imagine this in my mind for a long time. Curl has always daunted me in calculations, not mathematically but rather with it's physical interpretation. What I've been able to analyze so far is that, when we analyze the time variant magnetic field, it will produce the curl in the perpendicular direction of it's flow. And this curl will be J, current density as curling magnetic field always indicate presence of current.

Differential form of Faraday's law states that curl of electric field will yield time-changing magnetic flux density. this one is still in the process of being imagined. As changing electric field will certainly produce changing current and changing current will always have changing magnetic field around it.

Gauss law for electric field in differential form was easy to analyze as divergence of electric flux density, which is the measure of outflow-inflow, gives charge density.

And the differential form of Gauss law for magnetic fields states that divergence of magnetic flux density is equal to zero as magnetic field through a closed surface will always have equal inflow and outflow.

$\nabla \cdot \mathbf{D} = \rho_f$