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