Topic 3 Electromagnetic Waves 1 UEEP1033 Oscillations and Waves Topic 6 Electromagnetic Waves Types of electromagnetic waves Electromagnetic spectrum Propagation of electromagnetic wave Electric field and magnetic field Qualitative treatment of electromagnetic waves

Topic 3 Electromagnetic Waves 2 UEEP1033 Oscillations and Waves Electromagnetic (EM) waves were first postulated by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz Maxwell derived a wave form of the electric and magnetic equations, revealing the wave-like nature of electric and magnetic fields, and their symmetry Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave According to Maxwell’s equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on These oscillating fields together form an electromagnetic wave Introduction

Topic 3 Electromagnetic Waves 3 UEEP1033 Oscillations and Waves In the studies of electricity and magnetism, experimental physicists had determined two physical constants - the electric (  o ) and magnetic (  o ) constant in vacuum These two constants appeared in the EM wave equations, and Maxwell was able to calculate the velocity of the wave (i.e. the speed of light) in terms of the two constants: Therefore the three experimental constants,  o,  o and c previously thought to be independent are now related in a fixed and determined way Speed of EM waves  0 =  C 2 s 2 /kgm 3 (permittivity of vacuum)  0 = 4   kgm/A 2 s 2 (permeability of vacuum)

Topic 3 Electromagnetic Waves 4 UEEP1033 Oscillations and Waves NameDifferential formIntegral form Gauss's law Gauss's law for magnetism Maxwell–Faraday equation (Faraday's law of induction) Ampère's circuital law (with Maxwell's correction) Formulation in terms of free charge and current Maxwell’s Equations

Topic 3 Electromagnetic Waves 5 UEEP1033 Oscillations and Waves Maxwell’s Equations Formulation in terms of total charge and current Differential formIntegral form Gauss's law Gauss's law for magnetism Maxwell–Faraday equation (Faraday's law of induction) Ampère's circuital law (with Maxwell's correction)

Topic 3 Electromagnetic Waves 6 UEEP1033 Oscillations and Waves line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve) line integral of the magnetic field over the closed boundary ∂S of the surface S The electric flux (surface integral of the electric field) through the (closed) surface (the boundary of the volume V ) The magnetic flux (surface integral of the magnetic B-field) through the (closed) surface (the boundary of the volume V ) Maxwell’s Equations

Topic 3 Electromagnetic Waves 7 UEEP1033 Oscillations and Waves (1) Gauss’s law for the electric field Gauss’s law is a consequence of the inverse-square nature of Coulomb’s law for the electrical force interaction between point like charges (2) Gauss’s law for the magnetic field This statement about the non existence of magnetic monopole; magnets are dipolar. Magnetic field lines form closed contours (4) The Ampere-Maxwell law This law is a statement that magnetic fields are caused by electric conduction currents and or by a changing electric flux (via the displacement current) (3) Faraday’s law of electromagnetic induction This is a statement about how charges in magnetic flux produce (non-conservative) electric fields Maxwell’s Equations

Topic 3 Electromagnetic Waves 8 UEEP1033 Oscillations and Waves Electromagnetic Spectrum

Topic 3 Electromagnetic Waves 9 UEEP1033 Oscillations and Waves Generating an Electromagnetic Waves An arrangement for generating a traveling electromagnetic wave in the shortwave radio region of the spectrum: an LC oscillator produces a sinusoidal current in the antenna, which generate the wave. P is a distant point at which a detector can monitor the wave traveling past it

Topic 3 Electromagnetic Waves 10 UEEP1033 Oscillations and Waves Generating an Electromagnetic Waves Variation in the electric field E and the magnetic field B at the distant point P as one wavelength of the electromagnetic wave travels past it. The wave is traveling directly out of the page The two fields vary sinusoidally in magnitude and direction The electric and magnetic fields are always perpendicular to each other and to the direction of travel of the wave

Topic 3 Electromagnetic Waves 11 UEEP1033 Oscillations and Waves Close switch and current flows briefly. Sets up electric field Current flow sets up magnetic field as little circles around the wires Fields not instantaneous, but form in time Energy is stored in fields and cannot move infinitely fast Generating an Electromagnetic Waves

Topic 3 Electromagnetic Waves 12 UEEP1033 Oscillations and Waves Figure (a) shows first half cycle When current reverses in Figure (b), the fields reverse See the first disturbance moving outward These are the electromagnetic waves Generating an Electromagnetic Waves

Topic 3 Electromagnetic Waves 13 UEEP1033 Oscillations and Waves Notice that the electric and magnetic fields are at right angles to one another They are also perpendicular to the direction of motion of the wave Generating an Electromagnetic Waves

Topic 3 Electromagnetic Waves 14 UEEP1033 Oscillations and Waves Electromagnetic Waves The cross product always gives the direction of travel of the wave Assume that the EM wave is traveling toward P in the positive direction of an x-axis, that the electric field is oscillating parallel to the y-axis, and that the magnetic filed is the oscillating parallel to the z-axis: E 0 = amplitude of the electric field B 0 = amplitude of the magnetic field  = angular frequency of the wave k = angular wave number of the wave At any specified time and place: E/B = c (speed of electromagnetic wave)

Topic 3 Electromagnetic Waves 15 UEEP1033 Oscillations and Waves Electromagnetic wave represents the transmission of energy The energy density associated with the electric field in free space: The energy density associated with the magnetic field in free space: Electromagnetic Waves Total energy density:

Topic 3 Electromagnetic Waves 16 UEEP1033 Oscillations and Waves Example Imagine an electromagnetic plane wave in vacuum whose electric field (in SI units) is given by Determine (i) the speed, frequency, wavelength, period, initial phase and electric field amplitude and polarization, (ii) the magnetic field. Solution (i) The wave function has the form:

Topic 3 Electromagnetic Waves 17 UEEP1033 Oscillations and Waves Solution (continued) Period, and the initial phase = 0 Electric field amplitude The wave is linearly polarized in the x-direction and propagates along the z-axis (ii) The wave is propagating in the z-direction whereas the electric field oscillates along the x-axis, i.e. resides in the xz-plane. Now, is normal to both and z-axis, so it resides in the yz- plane. Thus, Since,

Topic 3 Electromagnetic Waves 18 UEEP1033 Oscillations and Waves refer to the fields of a wave at a particular point in space and indicates the Poynting vector at that point Energy Transport and the Poynting Vector Like any form of wave, an EM wave can transport from one location to another, e.g. light from a bulb and radiant heat from a fire The energy flow in an EM is measured in terms of the rate of energy flow per unit area The magnitude and direction of the energy flow is described in terms of a vector called the Poynting vector: is perpendicular to the plane formed by, the direction is determined by the right-hand rule.

Topic 3 Electromagnetic Waves 19 UEEP1033 Oscillations and Waves Energy Transport and the Poynting Vector Because are perpendicular to each other in an EM wave, the magnitude of is: Instantaneous energy flow rate Intensity I of the wave = time average of S, taken over one or more cycles of the wave In terms of rms :

Topic 3 Electromagnetic Waves 20 UEEP1033 Oscillations and Waves Example [source: Halliday, Resnick, Walker, Fundamentals of Physics 6 th Edition, Sample Problem 34-1 An observer is 1.8 m from a light source whose power P s is 250 W. Calculate the rms values of the electric and magnetic fields due to the source at the position of the observer. Energy Transport and the Poynting Vector

Topic 3 Electromagnetic Waves 21 UEEP1033 Oscillations and Waves Polarization of Electromagnetic Wave

Topic 3 Electromagnetic Waves 22 UEEP1033 Oscillations and Waves Polarization of Electromagnetic Wave The transverse EM wave is said to be polarized (more specifically, plane polarized) if the electric field vectors are parallel to a particular direction for all points in the wave direction of the electric field vector E = direction of polarization Example, consider an electric field propagating in the positive z-direction and polarized in the x-direction

Topic 3 Electromagnetic Waves 23 UEEP1033 Oscillations and Waves Example A plane electromagnetic harmonic wave of frequency 600  Hz, propagating in the positive x-direction in vacuum, has an electric field amplitude of V/m. The wave is linearly polarized such that the plane of vibration of the electric field is at 45 o to the xz-plane. Obtain the vector Solution here x y z

Topic 3 Electromagnetic Waves 24 UEEP1033 Oscillations and Waves Solution (continued) So

Topic 3 Electromagnetic Waves 25 UEEP1033 Oscillations and Waves Harmonic Waves A = amplitudek = 2  / (propagation constant) or v = f = f (2  /k)k v = 2  f =  (angular frequency) or Phase :  = k(x + vt) = kx +  t  moving in the – x-direction  = k(x - vt) = kx -  t  moving in the + x-direction

Topic 3 Electromagnetic Waves 26 UEEP1033 Oscillations and Waves Harmonic Waves In general, to accommodate any arbitrary initial displacement, some angle  0 must be added to the phase, e.g. Suppose the initial boundary conditions are such that y = y 0 when x = 0 and t = 0, then y = A sin  0 = y 0   0 = sin -1 (y 0 /A)

Topic 3 Electromagnetic Waves 27 UEEP1033 Oscillations and Waves Plane Waves The wave “displacement” or disturbance y at spatial coordinates (x, y, z):  Traveling wave moving along the +x-direction At fixed time, let take at t = 0: When x = constant, the phase  = kx = constant  the surface of constant phase are a family of planes perpendicular to the x-axis  these surfaces of constant phase are called the wavefronts

Topic 3 Electromagnetic Waves 28 UEEP1033 Oscillations and Waves Plane Waves Plane wave along x-axis. The waves penetrate the planes x = a, x = b, and, x = c at the points shown

Topic 3 Electromagnetic Waves 29 UEEP1033 Oscillations and Waves Plane Waves Generalization of the plane wave to an arbitrary direction. The wave direction is given by the vector k along the x-axis in (a) and an arbitrary direction in (b) x= r cos  are the components of the propagation direction

Topic 3 Electromagnetic Waves 30 UEEP1033 Oscillations and Waves Spherical & Cylindrical Waves Spherical Waves: Cylindrical Waves: r = radial distance from the point source to a given point on the waveform A/ r = amplitude  = perpendicular distance from the line of symmetry to a point on the waveform e.g. of the z-axis is the line of symmetry, then

Topic 3 Electromagnetic Waves 31 UEEP1033 Oscillations and Waves Mathematical Representation of Polarized Light Consider an EM wave propagating along the z-direction of the coordinate system shown in figure. The electric field of this wave at the origin of the axis system is given by: z x y 0 Propagation direction Complex field components for waves traveling in the +z-direction with amplitude E 0x and E 0y and phases  x and  y :

Topic 3 Electromagnetic Waves 32 UEEP1033 Oscillations and Waves = complex amplitude vector for the polarized wave Since the state of polarization of the light is completely determined by the relative amplitudes and phases of these components, we just concentrate only on the complex amplitude, written as a two- element matrix – called Jones vector: Mathematical Representation of Polarized Light

Topic 3 Electromagnetic Waves 33 UEEP1033 Oscillations and Waves Linear Polarization Figures representation of -vectors of linearly polarized light with various special orientations. The direction of the light is along the z-axis oscillations along the y-axis between +A and  A Vertically polarizedHorizontally polarizedLinearly polarized +A AA linear polarization along y

Topic 3 Electromagnetic Waves 34 UEEP1033 Oscillations and Waves Linear Polarization = Jones vector for vertically linearly polarized light = vector expression in normalized from for In general:

Topic 3 Electromagnetic Waves 35 UEEP1033 Oscillations and Waves Linear Polarization linear polarization along x Horizontally polarized +A-A-A

Topic 3 Electromagnetic Waves 36 UEEP1033 Oscillations and Waves Linear Polarization linear polarization at  oscillations along the a line making angle  with respect to the x-axis Linearly polarized

Topic 3 Electromagnetic Waves 37 UEEP1033 Oscillations and Waves Linear Polarization For example  = 60 o : Given a vector a, b = real numbers the inclination of the corresponding linearly polarized light is given by

Topic 3 Electromagnetic Waves 38 UEEP1033 Oscillations and Waves Suppose  = negative angle  E 0y = negative number Since the sine is an odd function, thus E 0x remain positive The negative sign ensures that the two vibrations are  out of phase, as needed to produce linearly polarized light with -vectors lying in the second and fourth quadrants The resultant vibration takes places place along a line with negative slope Jones vector with both a and b real numbers, not both zero, represents linearly polarized light at inclination angle Linear Polarization

Topic 3 Electromagnetic Waves 39 UEEP1033 Oscillations and Waves In determining the resultant vibration due to two perpendicular components, we are in fact determining the appropriate Lissajous figure If  other than 0 or , the resultant E-vector traces out an ellipse Lissajous Figures

Topic 3 Electromagnetic Waves 40 UEEP1033 Oscillations and Waves Lissajous figures as a function of relative phase for orthogonal vibrations of unequal amplitude. An angle lead greater than 180 o may also be represented as an angle lag of less that 180 o. For all figures we have adopted the phase lag convention  y  x Lissajous Figures

Topic 3 Electromagnetic Waves 41 UEEP1033 Oscillations and Waves Linear Polarization (  = m  )

Topic 3 Electromagnetic Waves 42 UEEP1033 Oscillations and Waves Circular Polarization (  =  /2)

Topic 3 Electromagnetic Waves 43 UEEP1033 Oscillations and Waves Elliptical Polarization