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Maxwell’s equations


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Maxwell’s equations

  1. 1. Maxwell’s Equations<br />Presented by :<br />Anup Kr Bordoloi<br />ECE Department ,Tezpur University<br />09-18-2008<br />
  2. 2. Contents<br /><ul><li>Field equations
  3. 3. Equation of continuity for time varying fields
  4. 4. Inconsistency of Ampere’s Law
  5. 5. Maxwell’s equations
  6. 6. Conditions at a Boundary surfaces</li></li></ul><li>The equations governing electric field due to charges at rest and the static magnetic field due to steady currents are<br />Contained in the above is the equation of continuity <br />Time Varying Fields:<br />From Faraday’s Law<br />In time varying electric and magnetic fields path of integration can be considered fixed. Faraday’s Law <br />becomes <br />Hence 1st equation becomes <br />
  7. 7. Equation of continuity for Time-Varying Fields:<br />From conservation of charge concept<br /> if the region is stationary<br />Divergence theorem<br /> time varying form of equation of <br />Continuity <br />Inconsistency of Ampere’s Law:<br />Taking divergence of Ampere’s law hence Ampere’s law is<br /> not consistent for time varying equation of continuity.<br /> (from Gauss’s Law)<br /> displacement current density.<br />
  8. 8. Hence Ampere’s law becomes .Now taking divergence results equation of continuity<br />Integrating over surface and applying Stokes’s theorem <br />magneto motive force around a closed path=total current enclosed by the path.<br />Maxwell’s equations: <br />These are electromagnetic equations .one form may be derived from the other with the help of Stokes’ theorem or the<br />divergence theorem <br />Contained in the above is the equation of continuity.<br />
  9. 9. Word statement of field equation:<br />1.The magneto motive force (magnetic voltage)around a closed path is equal to the conduction current<br />plus the time derivative of electric displacement through any surface bounded by the path.<br />2.The electromotive force (electric voltage)around a closed path is equal to the time derivative of <br />magnetic displacement through any surface bounded by the path<br />3.Total electric displacement through the surface enclosing a volume is equal to the total charge wihin<br /> the volume.<br />4.The net magnetic flux emerging through any closed surface is zero. <br />Interpretation of field equation:<br />Using Stokes’ theorem to Maxwell’s 2nd equation<br />Again from Faraday’s law region where there is no time varying magnetic flux ,voltage<br />around the loop would be zero the field is electrostatic and irrational.<br />Again <br /> there are no isolated magnetic poles or<br /> “magnetic charges” on which lines of magnetic flux can terminate(the lines of mag.flux are continuous)<br />
  10. 10. Boundary condition:<br />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<br />Discontinuity can be deduced from the Maxwell’s equations<br /> 1.<br /> over any closed surface S<br /> 2.<br /> for any surface S bounded by closed loop p<br />3.<br /><br /> 4.<br />From 1 <br />D1<br />a<br />1<br />2<br />D2<br />
  11. 11. The component of D that is perpendicular to the interface is discontinuous by an amount<br />lly from equation 2<br />From equation 3 <br />If width of the loop goes to zero,the flux vanishes.<br /> E parallel to the interface is continuous.<br />From equation 4 <br />Current passing through the amperian loop ,No volume current density will continue, but a surface current can.<br />But <br />1<br />l<br />2<br />
  12. 12. In case of linear media above<br />boundary conditions can be written as if there is no free charge or free current at the interface<br />
  13. 13. Condition at boundary surfaces:<br />Space derivative can’t yield information about the points of discontinuity in the medium. integral form <br /> can do the task.<br />From Maxwell’s 2nd equation<br />From the fig.<br />Area of the rectangle is made to approach to zero reducing it’s width <br />y<br />x<br />
  14. 14. Tangential component of E is continuous.<br />lly tangential component of H is continuous(for finite <br /> current density)<br />Condition for normal component of B and D:<br />Integral form of 3rd equation<br />For elementary pillbox<br /> for the case of no surface charge <br />For metallic surface if surface charge density the charge density of surface layer is<br />
  15. 15. For metallic conductor it is zero for electrostatic case or in the case of a perfect<br /> conductor <br /> normal component of the displacement density of <br /> dielectric =surface charge density of on the conductor.<br />Similar analysis leads for magnetic field <br />
  16. 16. Electromagnetic Waves in homogeneous medium:<br />The following field equation must be satisfied for solution of electromagnetic problem<br /> there are three constitutional relation which determines <br /> characteristic of the medium in which the fields exists.<br />Solution for free space condition:<br />in particular case of e.m. phenomena in free space or in a perfect dielectric containing no charge and<br /> no conduction current<br />Differentiating 1st<br />
  17. 17. Also since and are independent of time <br />Now the 1st equation becomes on differentiating it<br />Taking curl of 2nd equation <br /> ( )<br />But<br /> this is the law that E must obey .<br />lly for H<br /> these are wave equation so E and H satisfy wave <br /> equation.<br />
  18. 18. Uniform Plane wave propagation:<br />If E and H are considered to be independent of two dimensions say X and Y<br />For uniform wave propagation differential equation <br /> equation for voltage or current<br /> along a lossless transmission line. <br />General solution is of the form <br /> reflected wave.<br />Uniform Plane Wave: <br />Above equation is independent of Y and Z and is a function of x and t only .such a wave is uniform plan <br /> wave.<br /> the plan wave equation may be written as component of E <br />
  19. 19. For charge free region<br /> for uniform plane wave there<br />is no component in X direction be either zero, constant in time or increasing<br />uniformly with time .similar analysis holds for H . Uniform plane electromagnetic<br /> waves are transverse and have components in E and H only in the direction perpendicular to <br /> direction of propagation<br />Relation between E and H in a uniform plane wave:<br />For a plane uniform wave travelling in x direction <br />a)E and H are both independent of y and z <br />b)E and H have no x component<br />From Maxwell’s 1st equation <br />From Maxwell’s 2nd equation<br />
  20. 20. Comparing y and z terms from the above equations<br /> on solving finally we get <br />lly<br />Since<br />The ratio has the dimension of impedance or ohms , called characteristic impedance or intrinsic impedance of the <br /> (non conducting) medium. For space <br />
  21. 21. The relative orientation of E and H may be determined by taking their dot product and using above relation <br /> In a uniform plane wave ,E and H are at right angles to each other.<br /> electric field vector crossed into the magnetic field vector gives the direction in which the wave travells.<br />
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  23. 23. Thank you<br />
  24. 24. Boundary condition:<br />Electric field suffers discontinuity at a surface charge,so the magnetic field is<br /> discontinuous at a surface current. only tangential component that changes<br />From the integral equation<br />Applying to the figure =<br />As for tangential component Amperical loop running perpendicular<br /> to current <br />Component of B parallel to the surface & perpendicular to the current is discontinuous by an amount<br />Amperical loop running parallel to the current shows parallel component is continuous.<br />Summary<br /> this is pointing upward ,vector perpendicular to the surface.<br />
  25. 25. Like the scalar potential in electrostatic vector potential is continuous across any boundary<br /> .for the normal component is <br /> continuous, in the form <br />Derivative of inherits the discontinuity of B.<br />