1. Maxwell’s Equations Presented by : Anup Kr Bordoloi ECE Department ,Tezpur University 09-18-2008

3. Equation of continuity for time varying fields

4. Inconsistency of Ampere’s Law

5. Maxwell’s equations

7. Equation of continuity for Time-Varying Fields: From conservation of charge concept if the region is stationary Divergence theorem time varying form of equation of Continuity Inconsistency of Ampere’s Law: Taking divergence of Ampere’s law hence Ampere’s law is not consistent for time varying equation of continuity. (from Gauss’s Law) displacement current density.

8. Hence Ampere’s law becomes .Now taking divergence results equation of continuity Integrating over surface and applying Stokes’s theorem magneto motive force around a closed path=total current enclosed by the path. Maxwell’s equations: These are electromagnetic equations .one form may be derived from the other with the help of Stokes’ theorem or the divergence theorem Contained in the above is the equation of continuity.

9. Word statement of field equation: 1.The magneto motive force (magnetic voltage)around a closed path is equal to the conduction current plus the time derivative of electric displacement through any surface bounded by the path. 2.The electromotive force (electric voltage)around a closed path is equal to the time derivative of magnetic displacement through any surface bounded by the path 3.Total electric displacement through the surface enclosing a volume is equal to the total charge wihin the volume. 4.The net magnetic flux emerging through any closed surface is zero. Interpretation of field equation: Using Stokes’ theorem to Maxwell’s 2nd equation Again from Faraday’s law region where there is no time varying magnetic flux ,voltage around the loop would be zero the field is electrostatic and irrational. Again there are no isolated magnetic poles or “magnetic charges” on which lines of magnetic flux can terminate(the lines of mag.flux are continuous)

10. Boundary condition: 1. E,B,D and H will be discontinuous at a boundary between two different media or at surface that carries charge density σ and current density K Discontinuity can be deduced from the Maxwell’s equations 1. over any closed surface S 2. for any surface S bounded by closed loop p 3. br /> 4. From 1 D1 a 1 2 D2

11. The component of D that is perpendicular to the interface is discontinuous by an amount lly from equation 2 From equation 3 If width of the loop goes to zero,the flux vanishes. E parallel to the interface is continuous. From equation 4 Current passing through the amperian loop ,No volume current density will continue, but a surface current can. But 1 l 2

12. In case of linear media above boundary conditions can be written as if there is no free charge or free current at the interface

13. Condition at boundary surfaces: Space derivative can’t yield information about the points of discontinuity in the medium. integral form can do the task. From Maxwell’s 2nd equation From the fig. Area of the rectangle is made to approach to zero reducing it’s width y x

14. Tangential component of E is continuous. lly tangential component of H is continuous(for finite current density) Condition for normal component of B and D: Integral form of 3rd equation For elementary pillbox for the case of no surface charge For metallic surface if surface charge density the charge density of surface layer is

15. For metallic conductor it is zero for electrostatic case or in the case of a perfect conductor normal component of the displacement density of dielectric =surface charge density of on the conductor. Similar analysis leads for magnetic field

16. Electromagnetic Waves in homogeneous medium: The following field equation must be satisfied for solution of electromagnetic problem there are three constitutional relation which determines characteristic of the medium in which the fields exists. Solution for free space condition: in particular case of e.m. phenomena in free space or in a perfect dielectric containing no charge and no conduction current Differentiating 1st

17. Also since and are independent of time Now the 1st equation becomes on differentiating it Taking curl of 2nd equation ( ) But this is the law that E must obey . lly for H these are wave equation so E and H satisfy wave equation.

18. Uniform Plane wave propagation: If E and H are considered to be independent of two dimensions say X and Y For uniform wave propagation differential equation equation for voltage or current along a lossless transmission line. General solution is of the form reflected wave. Uniform Plane Wave: Above equation is independent of Y and Z and is a function of x and t only .such a wave is uniform plan wave. the plan wave equation may be written as component of E

19. For charge free region for uniform plane wave there is no component in X direction be either zero, constant in time or increasing uniformly with time .similar analysis holds for H . Uniform plane electromagnetic waves are transverse and have components in E and H only in the direction perpendicular to direction of propagation Relation between E and H in a uniform plane wave: For a plane uniform wave travelling in x direction a)E and H are both independent of y and z b)E and H have no x component From Maxwell’s 1st equation From Maxwell’s 2nd equation

20. Comparing y and z terms from the above equations on solving finally we get lly Since The ratio has the dimension of impedance or ohms , called characteristic impedance or intrinsic impedance of the (non conducting) medium. For space

21. The relative orientation of E and H may be determined by taking their dot product and using above relation In a uniform plane wave ,E and H are at right angles to each other. electric field vector crossed into the magnetic field vector gives the direction in which the wave travells.

23. Thank you

24. Boundary condition: Electric field suffers discontinuity at a surface charge,so the magnetic field is discontinuous at a surface current. only tangential component that changes From the integral equation Applying to the figure = As for tangential component Amperical loop running perpendicular to current Component of B parallel to the surface & perpendicular to the current is discontinuous by an amount Amperical loop running parallel to the current shows parallel component is continuous. Summary this is pointing upward ,vector perpendicular to the surface.

25. Like the scalar potential in electrostatic vector potential is continuous across any boundary .for the normal component is continuous, in the form Derivative of inherits the discontinuity of B.