Introduction to Maxwell Equations

Maxwell's equations has a great impact on Electromagnetic theory. Here we are going to discuss about These in details.

Maxwell Equations | Electromagnetic Theory

Electrodynamics Before Maxwell : 

The four fundamental equations of electrodynamics are given by,

    These four equations represented the state of electromagnetic theory before Maxwell. But a big inconsistency is encountered in these formulas. Maxwell removed all the inconsistency and incompleteness of the above four equations.

    Faraday's law shows that a time varying magnetic field can create an electric field. Now, we may expect that time varying electric field may also create magnetic field. Maxwell proved it to be true by Making the correction in Ampere's law and introducing the displacement current. In the next section we will discus about the incompleteness and the resolution of the this incompleteness.

Maxwell's Correction to Ampere's law :

•  The incompleteness in Ampere's Law :

        We Know that divergence of any vector is always zero.

            But, This is not consistent with The equation of continuity - 

    Which Indicates that , is not zero in general . ( is zero only for steady currents. )

    Let us take another example of Charging of capacitors ,

        When a Dc current is applied then the capacitor blocks the current and no current flows through the circuit and there is no problem. 

        But, when an ac current is applied then the capacitor allows the current through the circuit. Let's see what happens -   

    The integral form of Ampere's law is given by,

   

        In the above Example, when  we take the circular surface S1 as the amperian loop then the current enclosed in the loop is I and the left hand side is non zero. 

        But What will happen if we take the balloon shaped surface S2  as the amperian loop ? Is there any current enclosed by the loop ? 

        Simply, the answer is no because there is no wired connection puncturing the surface. How can the current flow without a conductor ?  i.e. there is no current enclosed by the loop. Then the left hand side is 0 . But how is it possible ? These two cases are contradictory. 

        As the Total current I flows through the whole circuit and the capacitor do not stop it then    there is something wrong in this case.

•   Resolution of the incompleteness :

        When the capacitor is being charged charges are pilling on the plates of the capacitor and this causes the charge density and electric field to vary with time. The energy is stored in the electric field. For the cases of such time varying fields, Ampere's law requires modification such that the law becomes consistent with the equation of continuity. Maxwell investigated the situation theoretically and suggested that the current density J (vector) in the amperes law is incomplete and we must replace it by  to make it consistent with the equation of continuity. then, 

----------------------(1)

[ According to the equation of continuity - 

       ]

Now, putting the value of ρ from the Gauss law in electrostatics - 

 - in the equation (1) we get,  

killing the divergence in both sides we get,

This term is known as displacement current. Displacement current is a current in the sense that it can produce a magnetic field. As it is not linked with the motion of free charges, it has none of the other properties of current.  

Thus, taking displacement current into account the corrected amperes law takes the form - 

 

Thus, finally the four Maxwell equations takes the form -  

Important Properties of Maxwell's Equations:  

• Maxwell equations are linear. It is directly related to the principle of superposition. Thus, if any two fields satisfy Maxwell's equations their sum will also satisfy Maxwell's equations.
• Maxwell 's equations also include the equation of continuity. Thus it also satisfies the local conservation of charge.
• Maxwell's equations are relativistic invariants i.e. They remain invariant under Lorentz Transformation. 
• Maxwell's equations are not symmetric with respect to electric and magnetic fields. This is due to the fact that electric charges exist in nature but magnetic charges do not exist. 
• Maxwell's equations predict the existence of electro magnetic waves because it shows that varying electric field creates magnetic field and vice-versa.