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# Electromagnetic waves

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### Electromagnetic waves

1. 1. Electromagnetic Waves<br />Presented by :<br />Anup Kr Bordoloi<br />ECE Department ,Tezpur University<br />11/11/2008<br />
2. 2. 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 exist.<br />Solution for free space condition:<br />in particular case of e.m. phenomena in free space or in a perfect dielectric containing no<br /> charge an no conduction current<br />Differentiating 1st<br />
3. 3. Also since and are independent of time <br />Now the 1st equation becomes on differentiating it<br />Taking curl of 2nd equation <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 equation.<br />
4. 4. For charge free region<br /> for uniform plane wave <br />There is no component in X direction be either zero, constant in<br /> time or increasing uniformly with time .similar analysis holds for H<br /> Uniform plane electromagnetic waves are transverse and have components in E and H only in the direction perpendicular to 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 />
5. 5. 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<br /> of the (non conducting) medium. For space <br />
6. 6. The relativeorientation 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 travels.<br />
7. 7. The wave equation for conducting medium:<br />From Maxwell’s equation if the medium has conductivity<br />Taking curl of 2nd eq. ( )<br />For any homogeneous medium in which is constant <br />But there is no net charge within a conductor<br />Hence wave equation for E.<br /> lly , wave equation for H.<br />Sinusoidal time variations: <br />where is the frequency of variation. <br /> time factor may be suppressed through the use phasor <br /> notation.<br />Time varying field may be expressed in terms of corresponding phasor quan<br />-tity <br />
8. 8. as<br />Phasor is defined by<br /> real<br />Phase is determined by of the <br /> complex number ,time varying <br /> field quantity may be expressed as<br />Maxwell’s equation in phasor form:<br /> for sinusoidal steady state we may substitute the phasor<br /> relation as <br />Imaginary axis<br />Real axis<br />
9. 9. which is the differential equation in phasor form.<br />Observation point:<br />Time varying quantity is replaced by phasor quantity<br />Time derivative is replaced with a factor <br />Maxwell’s equation becomes <br />The above equations contain the equation of continuity<br />The constitutive relation retain their forms <br />For sinusoidal time variations the wave equation for electric field in lossless medium<br />
10. 10. becomes<br />In a conducting medium the wave equation becomes <br />Wave propagation in lossless medium:<br />For uniform plane wave there is no variation w.r.t. Y or Z.<br />For Ey component solution may be written as<br />The time varying field is <br /> real <br />
11. 11. When c1 and c2 are real,<br />if c1 = c2 the two travelling waves combine to form standing wave which does not <br /> progress. <br />Wave velocity: if velocity is given by <br /> or<br /> phase –shift constant. <br />From fig.<br />Again <br />
12. 12. Thank you <br />