Chapter 9. Electromagnetic waves

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1 Chapter 9. lectromagnetic waves

2 9.1.1 The (classical or Mechanical) waves equation Given the initial shape of the string, what is the subsequent form, The displacement at point z, at the later time t, is the same as the displacement a distance vt to the left (i.e. at z - vt), back at time t = 0: It represents a wave of fixed shape traveling in the z direction at speed v. (O) (X)

3 (Classical) waves equation with a solution of the form: f (,) zt g( zt) Waves equation means a equation of motion governed by Newton s second law! (xample) Consider a stretched string which supports wave motion. The net transverse force on the segment between z and (z + z) is If the distortion of the string is not too great, sin ~ tan. If the mass per unit length is, Newton's second law says

4 9.1.3 oundary conditions: Reflection and Transmission For a sinusoidal incident wave, then, the net disturbance of the string is: At the join (z = 0), the displacement and slope just slightly to the left (z = 0 - ) must equal those slightly to the right (z = 0 + ), or else there would be a break between the two strings.

5 9.2 lectromagnetic waves in Vacuum In Vacuum, 0, J 0, q 0, I t (no free charges and no currents) 0 0 Let s derive the wave equation for and from the curl equations. t ach Cartesian component of and satisfies Notice the crucial role played by Maxwell's contribution to Ampere's law without it, the wave equation would not emerge, and there would be no electromagnetic theory of light.

6 Algebraic form of Maxwell's quations in free space i k. = 0 (i.e. is perpendicular to k) i k. = 0 (i.e. is perpendicular to k) i k = iω i k = i o μ o ω and are mutually perpendicular to each other, and are perpendicular to the direction of propagation of wave. y x k z

What is the relation between and? Or show that and are in same phase at any time in space. k t c c y z z y c o o when x k z y z y z y k k ut wt kx i o z wt kx i o y e e oth y and z are in same phase.

7 What is the relation between and? Or show that and are in same phase at any time in space. k t c c y z z y c o o when x k z y z y z y k k ut wt kx i o z wt kx i o y e e oth y and z are in same phase. Since is a real number, the electric and magnetic vectors are in phase; thus if at any instant, is zero then is also zero, when attains its maximum value, also attains its maximum value, etc. k / o o /c y z x c

Summary of Important Properties of lectromagnetic Waves The solutions (plane wave) of Maxwell s equations are wave-like with both and satisfying a wave equation.

8 Summary of Important Properties of lectromagnetic Waves The solutions (plane wave) of Maxwell s equations are wave-like with both and satisfying a wave equation. y z o o cos cos kx t kx t lectromagnetic waves travel through empty space with the speed of light c = 1/( 0 0 ) ½. The plane wave as represented by above is said to be linearly polarized because the electric vector is always along y-axis and, similarly, the magnetic vector is always along z-axis. z y x direction of propagation

9 The components of the electric and magnetic fields of plane M waves are perpendicular to each other and perpendicular to the direction of wave propagation. The latter property says that M waves are transverse waves. The magnitudes of and in empty space are related by / = c. o o k c oth and are at right angles to the direction of propagation. Thus the waves are transverse. k The electric and magnetic waves are interdependent; neither can exist without the other. Physically, an electric field varying in time produces a magnetic field varying in space and time; this changing magnetic field produces an electric field varying in space and time and so on. This mutual generation of electric and magnetic fields result in the propagation of the M waves.

10 Numerical example In free space the lectric field is given as 10sin (2x 100t) ˆ. j Determine D, and H by using Maxwell s equations. Sol: Wave is propagating along x direction. (1) (2) (3) D sin (2x 100t) ˆ. j Using, t 20 (2 100 ) ˆ Cos x t k, t 1 Sin(2x 100 t) kˆ 1. Sin (2x 100t) kˆ H Sin(2x 100t) kˆ z y k x

11 9.3 lectromagnetic waves in Matter In linear and homogeneous media with no free charge and no free current, t t oundary conditions I S S average

12 9.3.2 Reflection and Transmission at Normal Incidence Suppose xy plane forms the boundary between two linear media. A plane wave of frequency ω is traveling in the z direction (from left), polarized along x direction (T polarization). Transmitted wave Incident wave I (,) zt 0I expikz 1 tx 1 I (,) zt 0I expikz 1 ty v Reflected wave 1 R (,) z t 0R expik1ztx 1 R (,) z t 0R expik1zty v 2 1 Transmitted wave T (,) zt 0T expikz 2 tx 1 T (,) zt 0T expikz 2 ty v At z = 0, Prove! R + T = 1

13 9.3.3 Reflection and Transmission at Oblique Incidence x Transmitted wave ecause the boundary conditions must hold at all points on the plane, and for all times, the exponential factors must be equal at z = 0 plane. Incident wave if y = 0 Reflected wave Transmitted wave Prove!

14 9.4 Absorption and Dispersion lectromagnetic waves in Conductors According to Ohm's law, the (free) current density is proportional to the electric field: Maxwell' s equations for linear media with no free charge assume the form, Plane-wave solutions are complex wavenumber Prove! k 1i The imaginary part,, results in an attenuation of the wave (decreasing amplitude with increasing z): skin depth

15 Determine the relative amplitudes, phases, and polarizations of and in conductors For field polarized along the x direction, Let s express the complex wavenumber in terms of its modulus and phase and fields are no longer in phase field lags behind field. The (real) electric and magnetic fields are, finally,

16 @ Helmholtz equation (wave equation in temporal frequency-domain) 2 2 (,) rt (,) rt rt 2 (,) 0 t t Scalar wave equation in space-domain Let us consider the Fourier transform of the electromagnetic field: 2 2 (,) rt (,) rt (,) rt 2 t t 2 2 Helmholtz equation k 1 (r, ) 0 k k j j

17 @ Frequency-domain Maxwell equations in a source-free space Using the temporal inverse Fourier transform, D H J t y similar reasoning, finally we have the frequency-domain equations of H(r, ) J(r, ) jd(r, ) (r, ) j(r, ) D(r, ) (r, ) (r, ) 0 and J(r, ) j (r, ) The frequency-domain equations involve one fewer derivative (the time derivative has been replaced by multiplication by jω), hence may be easier to solve. However, the inverse transform may be difficult to compute.

18 9.4.2 Reflection at a conducting surface The general boundary conditions for electrodynamics; ( f : the free surface charge, K f : the free surface current) A monochromatic plane wave, traveling in z, polarized in x, approaches from the left, attenuated as it penetrates into the conductor Prove!

$2 d x Fnet Fbinding Fdamping Fdriving m dt 2 : dipole moment : polarization vector : complex permittivity : absorption coefficient : refractive$

19 9.4.3 Frequency dependence of permittivity in dielectric media (Dispersion) The electrons in a dielectric are bounded to specific molecules. 2 d x Fnet Fbinding Fdamping Fdriving m dt 2 : dipole moment : polarization vector : complex permittivity : absorption coefficient : refractive index

20 9.4.3 Frequency dependence of permittivity in dielectric media If you agree to stay away from the resonances, the damping can be ignored anomalous dispersion Cauchy's formula

CHAPTER 9 ELECTROMAGNETIC WAVES

CHAPTER 9 ELECTROMAGNETIC WAVES Outlines 1. Waves in one dimension 2. Electromagnetic Waves in Vacuum 3. Electromagnetic waves in Matter 4. Absorption and Dispersion 5. Guided Waves 2 Skip 9.1.1 and 9.1.2