ELECTROMAGNETIC WAVES AND PHOTONS

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1 CHAPTER ELECTROMAGNETIC WAVES AND PHOTONS Problem.1 Find the magnitude and direction of the induced electric field of Example.1 at r = 5.00 cm if the magnetic field change at a contant rate from T to zero in Aume that the cylindrical region ha a radiu of 10.0 cm. According the reult of Example.1 E = r db dt for a point inide the cylindrical region. Subtitution give E = 0.05m ) 0.5T = 0.15 V 0.1 m Since the magnetic field magnitude i decreaing, the induced electric field i azimuthal in the clockwie ene. On line verion ue a magnetic field that i out of the page with a magnitude that increae; thu, the induced electric field i clockwie elected from a drop down lit). The tarting value of the magnetic field i randomized: E = 0.05m ) Bi 0.1 Intructor Manual for Principle of Phyical Optic. By C. A. Bennett c 008 John Wiley & Son, Inc. 3

2 4 ELECTROMAGNETIC WAVES AND PHOTONS Problem. A cylindrical region of pace of radiu R contain a uniform magnetic field B with direction into the page, a hown in Figure.. If the magnitude B inide the cylinder change in time and outide the cylinder it i zero, decribe the induced electric field magnitude and direction) for point outide the cylinder r > R). Find the magnitude and direction of the induced electric field at r = cm if R = 10.0 cm and the magnetic field change at a contant rate from zero to T in Since r > R, Thu Subtitution give C E d l = E πr) = d dt E = 0.1m) 0.15m) A E = R db r dt B d A = πr db dt ) 0.5T = V 0.1 m Since B increae, the azimuthal ene i counter clockwie, a in Figure.. On line verion ue a magnetic field that point out of the page with increaing magnitude, o the azimuthal ene of the induced electric field i clockwie elected from a drop down lit). The problem contain help with the integration idea, ince the cae where r > R i not dicued in the text. The initial value of the magnetic field i randomized: E = 0.1m) 0.15m) Bi 0.1 ) Problem.3 Find the magnitude and direction of the induced magnetic field of Example. at r = 5.00 cm if the electric field change at a contant rate from 5000 V/m to zero in Aume that the cylindrical region ha a radiu of 10.0 cm. According the reult of Example. r de B = µ 0 ɛ 0 dt for a point inide the cylindrical region. Subtitution give ) ) m 5000 V m B = ) m = T 0.1 Since the electric field magnitude i decreaing, the induced electric field i azimuthal in the counter clockwie ene.

3 5 On line verionue an electric field that i out of the page with a magnitude that increae; thu, the induced electric field i counter clockwie elected from a drop down lit). The tarting value of the electric field i randomized: B = m ) 0.05 m ) ) Ei 0.1 Problem.4 A cylindrical region of empty pace of radiu R contain a uniform electric field E with direction into the page, a hown in Figure.4. If the magnitude E inide the cylinder change in time and outide the cylinder it i zero, decribe the induced magnetic field magnitude and direction) for point outide the cylinder r > R). Find the magnitude and direction of the induced magnetic field at r = cm if R = 10.0 cm and the electric field change at a contant rate from zero to 5000 V/m in Since r > R, Thu C Subtitution give B = B d d l = B πr) = µ 0 ɛ 0 dt m A B = µ 0 ɛ 0 R ) 0.1m) 0.15m) r E d A = µ 0 ɛ 0 πr de dt de dt Since E decreae, the azimuthal ene i counter clockwie. ) 5000 V m = T 0.1 On line verionue an electric field that i out of the page with a magnitude that increae; thu, the induced magnetic field i counter clockwie elected from a drop down lit). The tarting value of the electric field i randomized: B = m ) 0.1m) 0.15m) ) Ei C Problem.5 A region of pace ha a permittivity of Nm. What i the peed of electromagnetic radiation within thi region? What i the index of refraction of thi material? The relative permittivity i K E = =

4 6 ELECTROMAGNETIC WAVES AND PHOTONS The index of refraction i n = K E = 1.43 giving an electromagnetic wave peed of v = c n = m On line verion randomize the permittivity. Problem.6 Derive the differential wave equation for B Equation.17). Begin by taking the time derivative of Faraday law, then ue Ampere law. Ue Equation.A.5 for the x component, followed by Equation.A.7 and.a.6: B x t = Ez t y E ) y z = ) Ey ) Ez z t y t = 1 [ Bx ɛµ 0 z z B ] z 1 [ By x ɛµ 0 y x B ] x y = 1 [ ] B x ɛµ 0 z + B x 1 [ Bz y ɛµ 0 x z + B ] y y Add in and out the term B x x : B x t = 1 [ B x ɛµ 0 x + B x ] B x y z 1 [ Bx ɛµ 0 x x + B y y + B ] z z The lat term i zero by Gau law for B. Thu B x x + B x B x y z Similar equation follow for B y and B z. Thu B = ɛµ 0 B t = ɛµ 0 B x t Problem.7 Show that in a tranvere electromagnetic wave, the magnetic field i perpendicular to the propagation direction.

5 7 According to Gau law for B Equation.A.10) B x x + B y y + B z z = îk x B x + ĵ k y B y + ˆk k z B z = i k B = 0 Problem.8 A plane electromagnetic wave traveling in vacuum i decribed by ) Ex, y, z, t) = E 0 ĵ e ikz ωt) Find an equation for the magnetic field of thi wave. Thi i a forward traveling wave traveling along z with electric field that at thi intant point along +y. According to Poynting, the magnetic field mut point along x. Thu Bx, y, z, t) = E ) 0 c î e ikz ωt) On line verion randomize E 0, then ak for variou parameter of the magnetic field expreion. Problem.9 A plane electromagnetic wave traveling in vacuum i decribed by ) Bx, y, z, t) = B 0ˆk e ikx+ωt) Find an equation for the electric field of thi wave. The wave propagate along x with magnetic field that at thi intant point along +z. The correponding electric field mut point along y: ) Ex, y, z, t) = cb 0 ĵ e ikx+ωt) On line verion randomize B 0, then ak for variou parameter of the electric field expreion. Problem.10 A plane electromagnetic wave traveling in vacuum i decribed by ) Ex, y, z, t) = E 0 î e iky ωt) Find an equation for the magnetic field of thi wave.

6 8 ELECTROMAGNETIC WAVES AND PHOTONS The wave propagate along +y with electric field that at thi intant point along x. The correponding magnetic field mut point along +z: ) E0 Bx, y, z, t) = c ˆk e iky ωt) On line verion randomize E 0, then ak for variou parameter of the magnetic field expreion. ) Problem.11 Show that average of in k r ωt + ϕ over many cycle i 1/. Work a in Example.4, uing the trigonometric identity in θ = 1 1 co θ) Problem.1 Let E = 00 V ) iky ω t) ˆk e m with ω = rad/ and k = rad/m. Find the correponding magnetic field and the irradiance of the wave. The wave peed i v = ω k = 108m Thu, the index of refraction i n =.00. The wave propagate along +y with electric field that at thi intant point along +z. The correponding magnetic field mut point along +x: ik y ω t) Bx, y, z, t) = B 0 îe where the magnetic field magnitude i B 0 = E 0 v = 00 V m 10 8 m = 10 6 T Find the irradiance a in Example.5, or ue ) I = nɛ 1 C 3.00 ).00) c E 0 = n m 10 8 m On line verion randomize E 0, ω and k. 00 V ) = 107 W m m

7 9 Problem.13 Let B = T ) ik z ω t) ĵ e with ω = rad/ and k = rad/m. Find the correponding electric field and the irradiance of the wave. The wave peed i v = ω k = 108m Thu, the index of refraction i n =.00. The wave propagate along +z with magnetic field that at thi intant point along +y. The correponding electric field mut point along +x: ikz ω t) Ex, y, z, t) = E 0 îe where the electric field magnitude i E 0 = v B 0 = m ) T ) = 180 V m Find the irradiance a in Example.5, or ue ) I = nɛ 1 C 3.00 ).00) c E 0 = n m 10 8 m 180 V ) = 86.1 W m m On line verion randomize B 0, ω and k. The initial magnetic field amplitude i negative, and the wave travel along x; thu the initial E 0 point along +z. Problem.14 Find the irradiance of the following tranvere electromagnetic wave, auming that they ) travel in vacuum: a) E = 00 V m î e ikz ωt) ) b) B = T ĵ e ikz ωt) a) I = ɛ 0c E 0 = ) 1 C 3.00 ) n m 10 8 m 00 V ) = 53.1 W m m b) I = ɛ 0c E 0 = E 0 = On line verion randomize both field m ) T ) = 10 V m ) 1 C 3.00 ) n m 10 8 m 10 V ) = 58.6 W m m

8 10 ELECTROMAGNETIC WAVES AND PHOTONS Problem.15 The output of a green laer pointer ha a beam power of 10.0 mw and a beam diameter of 1.00 mm. Calculate the beam irradiance, and the maximum value of electric and magnetic field within the beam. Aume uniform irradiance acro the beam cro ection. The beam i cylindrical, o I = P πr = 10 W π m) = W m Thu I E 0 = ɛ 0 c = ) W m ) ) = 309 V C Nm m m B 0 = E 0 c = T = T On line verion randomize the beam power. Problem.16 The output of a certain laer pointer conit of a beam with uniform irradiance acro the beam cro ection. Aume a beam power of 1.00 mw and a beam diameter of 1.00 mm. a) Calculate the beam irradiance. b) How many photon per econd pa by any point in the beam if the wavelength i 633 nm? c) Repeat part b) if the wavelength i 530 nm. a) The beam i cylindrical, thu b) Photon energy: Photon rate: hf = hc λ = 10 3 W I = π m) = 173 W m J ) m m r p = 10 3 W J = ) = J c) Photon energy goe up J) o photon rate ) goe down. On line verion randomize the beam power.

9 11 Problem.17 Calculate the radiation preure a the beam of Problem.16 impinge on perfectly reflecting and perfectly aborbing urface. Calculate the photon momentum in each cae. Ue the intenity calculated in Problem.16. Shiny urface: P = I c = ) 173 W m m = N m The aborbing urface give half of thi preure. When λ = 633 nm, the photon momentum i p = h λ = J m When λ = 530 nm, the photon momentum i p = h λ = J m On line verion randomize the beam power. = kg m = kg m Problem.18 An elliptically polarized electromagnetic wave ha perpendicular component of E that are out of phae. For example, E x = E 0x e ikz ωt) E y = E 0y e ikz ωt+ϕ) with E 0x and E 0y real. Find the magnetic field component of thi wave, auming that it travel in vacuum. Each component mut atify the Poynting relation. Thu B y = E 0x c eikz ωt) B x = E 0y c eikz ωt+φ) On line verion randomize the electric field amplitude. Problem.19 Show explicitly that B x, y, z, t) = B 0 e i k r ω t+ϕ) i a olution to the differential wave equation for B. In Carteian coordinate, B x = ik x B

10 1 ELECTROMAGNETIC WAVES AND PHOTONS B x = k x B B = k B B t = ω B The wave equation i atified provided v = ω k Problem.0 For an electromagnetic wave, how that k B = 0, and thu that B i perpendicular to the direction of propagation. Ue Gau law for B Equation.A.10): B x x + B y y + B z z = ik xb x + k y B y + k z B z ) = i k B = 0 Problem.1 For an electromagnetic wave, how that k E = ω B. k E = îky E z k z E y ) + ĵ k z E x k x E z ) + ˆk k x E y k y E x ) The z component of the expreion above wa verified in the text for a forward traveling wave. Equation.A.5 verifie the x component E z y E y z = ik ye z k z E y ) = iωb x Equation.A.4 verifie the y component in a imilar way. Problem. For an electromagnetic wave, how that k B = ɛ µ 0 ω E. k B = îk y B z k z B y ) + ĵ k z B x k x B z ) + ˆk k x B y k y B x ) Aume a forward traveling wave. By Equation.A.8 Ampere law): B z y B y z = ik ye z k z E y ) = µ 0 ɛ iωe x )

11 13 which verifie the x component. Similarly, Equation.A.7 and.a.6 verify the y and z component. Problem.3 Let fx, t) = f 0 e ikx ωt) with f 0 a real contant. a) Find Re[f ]. b) Find Re[f]). c) Show that Re[f]) = 1 f 0 + Re[f ] ). a) b) c) Re [ f ] = f0 co kx ωt) Re f ) = f 0 1 f 0 + Re [ f ]) = 1 f 0 + f0 co kx ωt)) = f0 co kx ωt) = Re f ) ) ) Problem.4 Show that the average of in k r ω t + ϕ and co k r ω t + ϕ over many cycle i zero. Work a in Example.4. Without the quare, both integral evaluate to zero. On line verion Problem.5 Show that the average of ) ) in k r ω t + ϕ co k r ω t + ϕ over many cycle i zero. Proceed a in Example.4. Ue the trigonometric identity in θ coθ = 1 in θ

12 14 ELECTROMAGNETIC WAVES AND PHOTONS Problem.6 Show that an electromagnetic wave with pherical wavefront of radiu r ha an irradiance that varie a 1/r. The irradiance varie a the amplitude quared. According to Section 1.7., a pherical electromagnetic wave ha amplitude proportional to 1/r. Problem.7 Calculate the peak electric and magnetic field 1.0 km from a 1.0 MW radio tation, auming that it radiate electromagnetic wave a an iotropic point ource. The wavefront are pherical, o the irradiance from point ource of power P i Thu with On line verion randomize r and P. I = P 4πr P E 0 = πr ɛ 0 c B 0 = E 0 c Problem.8 It i often convenient to define the optical thickne of a material a nd where n i the index of refraction and d i the phyical thickne. a) Find the optical thickne for vacuum and gla n = 1.5) for a phyical thickne of one meter. b) Find the time it take light to travel d = 1.0 m in vacuum and in gla of index For a phyical ditance of one meter, the optical thickne i 1.5 m when the index i 1.5, and i equal to the phyical ditance in vacuum. Light travel time are obtained from the optical thickne. On line verion randomize the index. Problem.9 The maximum electric field utainable in a material before electrical breakdown i called the dielectric trength. For dry air at STP, the dielectric trength i about V/m. a) Ue thi to etimate the maximum irradiance of a laer beam that can propagate through air. b) If the beam profile i uniform and the beam diameter i 10 cm, what i the maximum beam power if it i to travel through air?

13 15 c) What radiation preure would thi beam exert on aborbing and reflecting urface? In each cae, what net force would be exerted by a 10 cm diameter beam? The maximum irradiance i determined by the dielectric trength: I max = ɛ 0c E 0 = The beam i cylindrical, o ) 1 C 3 ) Nm 10 8 m N ) = W C m I = P b πr If r = 5 cm, the maximum beam power i P b,max = πr I max = W In thi cae, the radiation preure exerted on a hiny urface i P max = I max c = 79.7 N m and half of thi for an aborbing urface. The net force exerted by a 10.0 cm diameter beam illuminating a hiny urface i 0.66 N and half of that for an aborbing urface. On line verion randomize the beam diameter. Problem.30 A certain puled laer ha a maximum power output of 10 7 W. a) Find the maximum value of E and B if the beam diameter i 10 cm. Aume a uniform beam profile. b) Find the maximum value of E and B if the beam i focued to a pot of diameter 100 µm. The beam cro ection i circular, o Thu, I = P b πr = ɛ 0c E 0 E 0 = Pb ɛ 0 cπr If r = 0.05 m, E 0 = V m. If r = m, E 0 = V m. In each cae, B 0 = E 0 /c. On line verion randomize the beam power. Problem.31 Calculate the electric and magnetic field at the top of Earth atmophere where the olar irradiance i 1340 W/m.

14 16 ELECTROMAGNETIC WAVES AND PHOTONS with B 0 = E 0 /c. On line verion i not randomized. I = ɛ 0c I E 0 E 0 = ɛ 0 c = 1004V m Problem.3 Find the frequency of a photon whoe momentum i that of a g BB traveling at 100 m/. The BB momentum i p = mv = kg m/. The correponding photon wavelength i λ = h/p, and the frequency i f = c/λ = Hz. On line verion randomize the BB velocity and ma. Problem.33 Determine the reponivity at 550 nm of a photomultiplier tube that ha a gain of 10 6 and a quantum efficiency of 30%. Repeat for a phototube with unity gain and the ame quantum efficiency. Thi give a reponivity of R = Gne hf = Gneλ hc A W for unity gain, and 1, A W for a gain of On line verion randomize the gain and quantum efficiency. Problem.34 A 100 W laer beam with.00 mm beam diameter illuminate a urface that aborb 40% and reflect 60%. Find the net force on the urface due to radiation preure. The radiation preure i P = 0.4I) c + 0.6I) c = 1.8 I c

15 17 The beam i cylindrical, o I = P b πr The force i F = P b πr = 1.8P b c = W = N c On line verion randomize the percentage aborbed and reflected. Problem.35 What i the minimum detectable wavelength of a photodetector with a cathode workfunction of.6 ev? Set the photon energy equal to the work function: W = hf = hc λ λ = hc W = J ) ) m.6ev ) ) J = 519nm ev On line verion randomize the work function. Problem.36 Under ideal circumtance, the human eye can detect a photon flux of about 50 photon/ of 550 nm light. If the eye ha a pupil diameter of 8.0 mm, what irradiance doe thi correpond to? The photon energy i E = hc λ = J The correponding irradiance i I = P b πr = 50 1 ) J ) On line verion randomize the pupil diameter. π m) = W m

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