Magnetism6.pptx

Lecture 2.6 Maxwell Equations in Matter:

When we have electric or magnetic polarization of matter then we have the bound charges and currents. The volume charge density and current density are:

----------(1) Electric Polarization ---------(2)Magnetization

Now in case of non-static we have the flow of polarization charge which is known as bound charge and this give rise to Polarization current and is given as

------------------(3) Note that this polarization current is different from the bound current density as latter is due to magnetization

-----------(4) Which is continuity eqn for the polarization of current

Now total charge density is written as sum of free charge density and bound charge density ------------(5) And total current density is written as sum of three terms --------- (6) Now applying the Gauss’s Law, we get, using (5)

1st Maxwell Eqn in Matter

------------------(7)

Where Displacement vector,

Now from Ampere’s modified Law we have ---------(8) Now using eqn (6), we get --------(9) Above eqn further can be written as -----------(10)

Where 4th Maxwell Eqn in matter ----------------(11)

Four Maxwell Eqns in Matter

Few Relations  



Displacement current

Wave Equations for Electric and Magnetic fields

We start with following Maxwell Eqns in space where no charge or current is,

Note that above four eqns are coupled 1st order Partial differential eqns.

Now we will decouple these. It means we write an eqn for E.F. which will not involve M.F. and an eqn for M.F. Which will not involve E.F. For this I will apply the curl to Eqn (iii) and (iv).

Applying curl to Eqn (iii), we have

----------(1) Where eqn (iv) is used Applying curl to Eqn (iv), we have

---------(2) Where eqn (iii) is used

Now using eqns, in equation (1) and (2) , we get --------(3)

Qbove eqn give us the wave eqn for E and B which Are 2nd order partial differential eqns. Thus we have now separate eqns for E and B

In general above equations look like

Where v is the propagation speed for electromagnetic waves in vacuum

Note that speed is same as that of light in vacuum

. Electromagnetic waves are transverse in nature i.e. E and B vectors are perpendicular to each other and also perpendicular to the direction of propagation. Exercise: Find the expressions for electromagnetic waves in conducting medium.