PHYS 1444 Section 3 Lecture #3 Monday, Nov. 8, 5 EM Waves from Maxwell s Equations Speed of EM Waves Light as EM Wave Electromagnetic Spectrum Energy in EM Waves Energy Transport The epilogue Today s homework is homework #1, noon, next Tuesday, Dec. 6!! Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 1
Reading assignments CH. 3 8 and 3 9 Announcements No class this Wednesday, Nov. 3 Final term exam Time: 11am 1:3pm, Monday Dec. 5 Location: SH13 Covers: CH 9.3 CH3 Please do not miss the exam Two best of the three exams will be used for your grades Monday, Nov. 8, 5 PHYS 1444-3, Fall 5
Maxwell s Equations In the absence of dielectric or magnetic materials, the four equations developed by Maxwell are: Q E da encl ε B da dφb E dl B dl µ I + µ ε encl dφ Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 Gauss Law for electricity A generalized form of Coulomb s law relating electric field to its sources, the electric charge Gauss Law for magnetism A magnetic equivalent ff Coulomb s law relating magnetic field to its sources. This says there are no magnetic monopoles. E Faraday s Law An electric field is produced by a changing magnetic field Ampére s Law A magnetic field is produced by an electric current or by a changing electric field 3
EM Waves and Their Speeds Let s consider a region of free space. What s a free space? An area of space where there is no charges or conduction currents In other words, far from emf sources so that the wave fronts are essentially flat or not distorted over a reasonable area What are these flat waves called? Plane waves At any instance E and B are uniform over a large plane perpendicular to the direction of propagation So we can also assume that the wave is traveling in the x- direction w/ velocity, vvi, and that E is parallel to y axis and B is parallel to z axis Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 4 v
Maxwell s Equations w/ QΙ In this region of free space, Q and Ι, thus the four Maxwell s equations become E da ε B da E dl Q encl dφ B dl µ I + µ ε B encl dφ Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 Q encl No Changes No Changes E I encl E da B da E dl dφ B dl µε One can observe the symmetry between electricity and magnetism. B dφ The last equation is the most important one for EM waves and their propagation!! E 5
EM Waves from Maxwell s Equations If the wave is sinusoidal w/ wavelength λ and frequency f, such traveling wave can be written as E E y E ( ) sin kx ωt B B B ( ) sin kx ωt Where z π k ϖ ω π f Thus f λ v λ k What is v? It is the speed of the traveling wave What are E and B? The amplitudes of the EM wave. Maximum values of E and B field strengths. Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 6
Let s apply Faraday s law From Faraday s Law dφ B E dl to the rectangular loop of height y and wih dx E dl Since E is perpendicular to dl. So the result of the integral through the loop counterclockwise becomes E dl E dx + ( ) E + de y+ E dx' + E y' + ( E + de) y E y de y For the right-hand side of Faraday s law, the magnetic flux through the loop changes as db de y dφ Thus dx y B db de dx Since E and B E dx y db depend on x and t x along the top and bottom of the loop is. Why? Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 7 B t
From Modified Ampére s Law Let s apply Maxwell s 4 th equation dφ E B dl µε to the rectangular loop of length z and wih dx B dl along the x-axis of the loop is Since B is perpendicular to dl. So the result of the integral through the loop counterclockwise becomes B dl B Z ( ) For the right-hand side of the equation is dφ de de E µε µε Thus db z dx z µε dx z db dx de B E µ ε Since E and B µε depend on x and t x t Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 B+ db Z db Z 8
Relationship between E, B and v Let s now use the relationship from Faraday s law Taking the derivatives of E and B as given their traveling wave form, we obtain E E ( kx ω t ) ke cos ( kx ω t) x x B t t Since ( ) sin E ( ω ) ( B kx t ) E B x t Thus sin We obtain ω v k Since E and B are in phase, we can write This is valid at any point and time in space. What is v? The velocity of the wave Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 E B x t ω B cos ( kx ωt ) cos ( ω ) ωb cos ( kx ωt ) ke kx t B E B v 9
Speed of EM Waves Let s now use the relationship from Apmere s law Taking the derivatives of E and B as given their traveling wave form, we obtain B ( B sin ( kx ω t )) kb cos ( kx ω t) x E t x t Since B x ( E sin ( kx ω t )) E ε µ t Thus B E ω E kx ωt cos ( ) However, from the previous page we obtain 1 Thus v 1 1 ε µ v εµ 1 7 ( 8.85 1 C N m ) ( 4 π 1 T m A) Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 B E ε µ x t We obtain kb cos ( kx ω t) ε µω E cos ( kx ωt ) ε µω ε µ k v E B 1 v ε µ v 3. 1 The speed of EM waves is the same as the speed of light. EM waves behaves like the light. 1 8 m s
Speed of Light w/o Sinusoidal Wave Forms Taking the time derivative on the relationship from Ampere s laws, we obtain B E ε µ xt t By the same token, we take position derivative on the E relationship from Faraday s law B x xt From these, we obtain E t 1 E εµ x and B t x Since the equation for traveling wave is v 1 t x By correspondence, we obtain v ε µ A natural outcome of Maxwell s equations is that E and B obey the wave equation for waves traveling w/ speed Maxwell predicted the existence of EM waves based on this 1 B εµ x x v 1 ε µ Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 11
Light as EM Wave People knew some 6 years before Maxwell that light behaves like a wave, but They did not know what kind of waves they are. Most importantly what is it that oscillates in light? Heinrich Hertz first generated and detected EM waves experimentally in 1887 using a spark gap apparatus Charge was rushed back and forth in a short period of time, generating waves with frequency about 1 9 Hz (these are called radio waves) He detected using a loop of wire in which an emf was produced when a changing magnetic field passed through These waves were later shown to travel at the speed of light Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 1
Light as EM Wave The wavelengths of visible light were measured in the first decade of the 19 th century The visible light wave length were found to be between 4.x1-7 m (4nm) and 7.5x1-7 m (75nm) The frequency of visible light is fλc Where f and λ are the frequency and the wavelength of the wave What is the range of visible light frequency? 4.x1 14 Hz to 7.5x1 14 Hz c is 3x1 8 m/s, the speed of light EM Waves, or EM radiation, are categorized using EM spectrum Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 13
Electromagnetic Spectrum Low frequency waves, such as radio waves or microwaves can be easily produced using electronic devices Higher frequency waves are produced natural processes, such as emission from atoms, molecules or nuclei Or they can be produced from acceleration of charged particles Infrared radiation (IR) is mainly responsible for the heating effect of the Sun The Sun emits visible lights, IR and UV The molecules of our skin resonate at infrared frequencies so IR is preferentially absorbed and thus warm up Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 14
Example 3 Wavelength of EM waves. Calculate the wavelength (a) of a 6-Hz EM wave, (b) of a 93.3-MHz FM radio wave and (c) of a beam of visible red light from a laser at frequency 4.74x1 14 Hz. What is the relationship between speed of light, frequency and the wavelength? c f λ Thus, we obtain For f6hz For f93.3mhz For f4.74x1 14 Hz λ c f 8 3 1 ms 6 λ 5 1 m 1 6s 8 3 1 ms λ 6 1 93.3 1 s 3.m 8 3 1 ms λ 14 1 4.74 1 s 6.33 1 Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 7 m 15
EM Wave in the Transmission Lines Can EM waves travel through a wire? Can it not just travel through the empty space? Nope. It sure can travel through a wire. When a source of emf is connected to a transmission line, the electric field within the wire does not set up immediately at all points along the line When two wires are separated via air, the EM wave travel through the air at the speed of light, c. However, through medium w/ permittivity e and permeability m, the speed of the EM wave is given v 1 εµ < c Is this faster than c? Nope! It is slower. Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 16
Energy in EM Waves Since BE/c and c 1 ε µ, we can rewrite the energy density 1 1 εµ E u u ε u εe E + µ ε E E + ub Note that the energy density associate with B field is the same as that associate with E So each field contribute half to the total energy By rewriting in B field only, we obtain 1 B 1 B B u ε + ε µ µ µ We can also rewrite to contain both E and B u ε E ε EcB EB εε µ ε EB µ Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 u u B µ ε µ 17 EB
Energy Transport What is the energy the wave transport per unit time per unit area? This is given by the vector S, the Poynting vector The unit of S is W/m. The direction of S is the direction in which the energy is transported. Which direction is this? The direction the wave is moving Let s consider a wave passing through an area A perpendicular to the x-axis, the axis of propagation How much does the wave move in time? dxc The energy that passes through A in time is the energy that occupies the volume dv, dv Adx Ac Since the energy density is uε E, the total energy, du, contained in the volume V is du udv ε E Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 Ac 18
Energy Transport Thus, the energy crossing the area A per time is 1 du S εce A Since EcB and c 1 ε µ, we can also rewrite S cb EB εce µ µ Since the direction of S is along v, perpendicular to E and B, the Poynting vector S can be written 1 S ( E B) µ This gives the energy transported per unit area per unit time at any instant Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 19
Average Energy Transport The average energy transport in an extended period of time since the frequency is so high we do not detect the rapid variation with respect to time. If E and B are sinusoidal, E E Thus we can write the magnitude of the average Poynting vector as 1 1 c EB S εce B This time averaged value of S is the intensity, defined as the average power transferred across unit area. E and B are maximum values. We can also write S E rms µ B rms µ µ Where Erms and Brms are the rms values ( E E, B ) rms B rms Monday, Nov. 8, 5 PHYS 1444-3, Fall 5
Example 3 4 E and B from the Sun. Radiation from the Sun reaches the Earth (above the atmosphere) at a rate of about 135W/m. Assume that this is a single EM wave and calculate the maximum values of E and B. What is given in the problem? The average S!! 1 1 c E S ε ce B B µ µ For E, E S ε c 135 W m ( 1 ) ( 8 8.85 1 C N m 3. 1 m s ) 1.1 1 3 V m For B B E 3 c 1.1 1 V m 8 3 1 ms 3.37 1 6 T Monday, Nov. 8, 5 PHYS 1444-3, Fall 5 1
You have worked very hard and well!! This was one of my best semesters!! Good luck with your final exams!! Have a safe winter break! Monday, Nov. 8, 5 PHYS 1444-3, Fall 5