ࡱ> qs/nopfJ  Y #`  7bjbj yͤ`f f f f > > > $b F F FPZFdKb LvLr(t:"u"u"uvvv$Zh> xv"vxxf f "u"uxf "u> "uxb$r > "ujL `P FZ}&VQd0> vUwhwTxvvv>^vvvxxxxb b b 3?$b b b ?b b b f f f f f f  Electromagnetic Waves 32.1. Identify: Since the speed is constant, distance  EMBED Equation.DSMT4  Set Up: The speed of light is  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  Evaluate: The speed of light is very great. The distance between stars is very large compared to terrestrial distances. 32.2. Identify: Since the speed is constant the difference in distance is  EMBED Equation.DSMT4  Set Up: The speed of electromagnetic waves in air is  EMBED Equation.DSMT4  Execute: A total time difference of  EMBED Equation.DSMT4 corresponds to a difference in distance of  EMBED Equation.DSMT4  Evaluate: The time delay doesn t depend on the distance from the transmitter to the receiver, it just depends on the difference in the length of the two paths. 32.3. Identify: Apply EMBED Equation.DSMT4  Set Up:  EMBED Equation.DSMT4  Execute: (a) EMBED Equation.DSMT4  (b) EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d) EMBED Equation.DSMT4  Evaluate: f increases when  EMBED Equation.DSMT4 decreases. 32.4. Identify:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Set Up:  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4 . UVA:  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . UVB:  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . (b)  EMBED Equation.DSMT4 . UVA:  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . UVB:  EMBED Equation.DSMT4 to  EMBED Equation.DSMT4 . Evaluate: Larger  EMBED Equation.DSMT4 corresponds to smaller f and k. 32.5. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Set Up: Since the wave is traveling in empty space, its wave speed is  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate: The  EMBED Equation.DSMT4 factor is common to both the electric and magnetic field expressions, since these two fields are in phase. 32.6. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Apply Eqs.(32.17) and (32.19). Set Up: The speed of the wave is  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , so that  EMBED Equation.DSMT4 will be in the  EMBED Equation.DSMT4 direction.  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Evaluate: The directions of  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 and of the propagation of the wave are all mutually perpendicular. The argument of the cosine is  EMBED Equation.DSMT4 since the wave is traveling in the  EMBED Equation.DSMT4 . Waves for visible light have very high frequencies. 32.7. Identify and Set Up: The equations are of the form of Eqs.(32.17), with x replaced by z.  EMBED Equation.DSMT4  is along the y-axis; deduce the direction of  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  is along the y-axis.  EMBED Equation.DSMT4  is in the direction of propagation (the +z-direction). From this we can deduce the direction of  EMBED Equation.DSMT4  as shown in Figure 32.7.  INCLUDEPICTURE FIG32-007.tif \* MERGEFORMAT \* MERGEFORMAT  EMBED Equation.DSMT4  is along the x-axis. Figure 32.7 EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4  Evaluate:  EMBED Equation.DSMT4  are perpendicular and oscillate in phase. 32.8. Identify: For an electromagnetic wave propagating in the negative x direction,  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Set Up: The wave specified in the problem has a different phase, so  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4 . (b)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . This wavelength is too short to be visible. (c)  EMBED Equation.DSMT4 . This is what the wave speed should be for an electromagnetic wave propagating in vacuum. Evaluate:  EMBED Equation.DSMT4  is an alternative expression for the wave speed. 32.9. Identify and Set Up: Compare the  EMBED Equation.DSMT4  given in the problem to the general form given by Eq.(32.17). Use the direction of propagation and of  EMBED Equation.DSMT4  to find the direction of  EMBED Equation.DSMT4  (a) Execute: The equation for the electric field contains the factor  EMBED Equation.DSMT4  so the wave is traveling in the +y-direction. The equation for  EMBED Equation.DSMT4  is in terms of  EMBED Equation.DSMT4  rather than  EMBED Equation.DSMT4  the wave is shifted in phase by  EMBED Equation.DSMT4  relative to one with a  EMBED Equation.DSMT4  factor. (b)  EMBED Equation.DSMT4  Comparing to Eq.(32.17) gives  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (c)  INCLUDEPICTURE FIG32-009.tif \* MERGEFORMAT \* MERGEFORMAT  EMBED Equation.DSMT4  must be in the +y-direction (the direction in which the wave is traveling). When  EMBED Equation.DSMT4  is in the z-direction then  EMBED Equation.DSMT4  must be in the x-direction, as shown in Figure 32.9. Figure 32.9 EMBED Equation.DSMT4   EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  Using Eq.(32.17) and the fact that  EMBED Equation.DSMT4  is in the  EMBED Equation.DSMT4  direction when  EMBED Equation.DSMT4  is in the  EMBED Equation.DSMT4  direction,  EMBED Equation.DSMT4  Evaluate:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are perpendicular and oscillate in phase. 32.10. Identify: Apply Eqs.(32.17) and (32.19).  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Set Up: The wave in this problem has a different phase, so  EMBED Equation.DSMT4  Execute: (a) The phase of the wave is given by  EMBED Equation.DSMT4 , so the wave is traveling in the  EMBED Equation.DSMT4  direction. (b)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  (c) Since the magnetic field is in the  EMBED Equation.DSMT4 -direction, and the wave is propagating in the  EMBED Equation.DSMT4 -direction, then the electric field is in the  EMBED Equation.DSMT4 -direction so that  EMBED Equation.DSMT4 will be in the  EMBED Equation.DSMT4 -direction.  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 have the same phase and are in perpendicular directions. 32.11. Identify and Set Up:  EMBED Equation.DSMT4  allows calculation of  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Eq.(32.18) relates the electric and magnetic field amplitudes. Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d) Eq.(32.18):  EMBED Equation.DSMT4  Evaluate: This wave has a very long wavelength; its frequency is in the AM radio braodcast band. The electric and magnetic fields in the wave are very weak. 32.12. Identify:  EMBED Equation.DSMT4  Set Up: The magnetic field of the earth is about  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4  Evaluate: The field is much smaller than the earth's field. 32.13. Identify and Set Up:  EMBED Equation.DSMT4  relates frequency and wavelength to the speed of the wave. Use Eq.(32.22) to calculate n and K. Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  Evaluate: In the material  EMBED Equation.DSMT4  is the same, so  EMBED Equation.DSMT4  is less in the material than in air.  EMBED Equation.DSMT4  always, so n is always greater than unity. 32.14. Identify: Apply Eq.(32.21).  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Apply Eq.(32.29) with  EMBED Equation.DSMT4 in place of  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  Evaluate: The wave travels slower in this material than in air. 32.15. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Set Up: The surface area of a sphere of radius r is  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4 . (b)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Evaluate: At the surface of the bulb the power radiated by the filament is spread over the surface of the bulb. Our calculation approximates the filament as a point source that radiates uniformly in all directions. 32.16. Identify and Set Up: The direction of propagation is given by  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  Evaluate: In each case the directions of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and the direction of propagation are all mutually perpendicular. 32.17. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  is in the direction of propagation. Set Up:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4 . For  EMBED Equation.DSMT4  in the +x-direction,  EMBED Equation.DSMT4  is in the +z-direction when  EMBED Equation.DSMT4  is in the +y-direction. Evaluate:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and the direction of propagation are all mutually perpendicular. 32.18. Identify: The intensity of the electromagnetic wave is given by Eq.(32.29):  EMBED Equation.DSMT4 The total energy passing through a window of area  EMBED Equation.DSMT4 during a time  EMBED Equation.DSMT4  is IAt. Set Up:  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4  Evaluate: The intensity is proportional to the square of the electric field amplitude. 32.19. Identify and Set Up: Use Eq.(32.29) to calculate I, Eq.(32.18) to calculate  EMBED Equation.DSMT4  and use  EMBED Equation.DSMT4  to calculate  EMBED Equation.DSMT4  (a) Execute:  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d) Evaluate: The calculation in part (c) assumes that the transmitter emits uniformly in all directions. 32.20. Identify and set up:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Execute: (a) The average power from the beam is  EMBED Equation.DSMT4 . (b)  EMBED Equation.DSMT4  Evaluate: The laser emits radiation only in the direction of the beam. 32.21: Identify:  EMBED Equation.DSMT4  Set Up: At a distance r from the star, the radiation from the star is spread over a spherical surface of area  EMBED Equation.DSMT4 . Execute:  EMBED Equation.DSMT4  Evaluate: The intensity decreases with distance from the star as  EMBED Equation.DSMT4 . 32.22. Identify and Set Up:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  Evaluate: Alternatively,  EMBED Equation.DSMT4 . 32.23. Identify:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Set Up: The surface area of a sphere is  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are both inversely proportional to the distance from the source. 32.24. Identify: The Poynting vector is  EMBED Equation.DSMT4  Set Up: The electric field is in the +y-direction, and the magnetic field is in the +z-direction.  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  The Poynting vector is in the  x-direction, which is the direction of propagation of the wave. (b)  EMBED Equation.DSMT4  But over one period, the cosine function averages to zero, so we have  EMBED Equation.DSMT4 . This is Eq.(32.29). Evaluate: We can also show that these two results also apply to the wave represented by Eq.(32.17). 32.25. Identify: Use the radiation pressure to find the intensity, and then  EMBED Equation.DSMT4  Set Up: For a perfectly absorbing surface,  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4 so  EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  Evaluate: Even though the source is very intense the radiation pressure 5.0 m from the surface is very small. 32.26. Identify: The intensity and the energy density of an electromagnetic wave depends on the amplitudes of the electric and magnetic fields. Set Up: Intensity is  EMBED Equation.DSMT4 , and the average power is Pav = 2I/c, where  EMBED Equation.DSMT4 . The energy density is  EMBED Equation.DSMT4  Execute: (a) I = Pav/A =  EMBED Equation.DSMT4  = 0.00201 W/m2. Pav = 2I/c =  EMBED Equation.DSMT4  = 1.34 ( EMBED Equation.DSMT4  Pa (b)  EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  = 1.23 N/C Bmax = Emax/c = (1.23 N/C)/(3.00 ( 108 m/s) = 4.10 (  EMBED Equation.DSMT4  T (c)  EMBED Equation.DSMT4  so  EMBED Equation.DSMT4  and Eav =  EMBED Equation.DSMT4 , so uav =  EMBED Equation.DSMT4  = 6.69 (  EMBED Equation.DSMT4  J/m3 (d) As was shown in Section 32.4, the energy density is the same for the electric and magnetic fields, so each one has 50% of the energy density. Evaluate: Compared to most laboratory fields, the electric and magnetic fields in ordinary radiowaves are extremely weak and carry very little energy. 32.27. Identify and Set Up: Use Eqs.(32.30) and (32.31). Execute: (a) By Eq.(32.30) the average momentum density is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (b) By Eq.(32.31) the average momentum flow rate per unit area is  EMBED Equation.DSMT4  Evaluate: The radiation pressure that the sunlight would exert on an absorbing or reflecting surface is very small. 32.28. Identify: Apply Eqs.(32.32) and (32.33). The average momentum density is given by Eq.(32.30), with S replaced by  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4  Execute: (a) Absorbed light:  EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  (b) Reflecting light:  EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  (c) The momentum density is  EMBED Equation.DSMT4  Evaluate: The factor of 2 in  EMBED Equation.DSMT4 for the reflecting surface arises because the momentum vector totally reverses direction upon reflection. Thus the change in momentum is twice the original momentum. 32.29. Identify: Apply Eq.(32.4) and (32.9). Set Up: Eq.(32.26) is  EMBED Equation.DSMT4 . Execute:  EMBED Equation.DSMT4  Evaluate: We can also write  EMBED Equation.DSMT4  S can be written solely in terms of E or solely in terms of B. 32.30. Identify: The electric field at the nodes is zero, so there is no force on a point charge placed at a node. Set Up: The location of the nodes is given by Eq.(32.36), where x is the distance from one of the planes.  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4  There must be nodes at the planes, which are 80.0 cm apart, and there are two nodes between the planes, each 20.0 cm from a plane. It is at 20 cm, 40 cm, and 60 cm from one plane that a point charge will remain at rest, since the electric fields there are zero. Evaluate: The magnetic field amplitude at these points isn t zero, but the magnetic field doesn t exert a force on a stationary charge. 32.31. Identify and Set Up: Apply Eqs.(32.36) and (32.37). Execute: (a) By Eq.(32.37) we see that the nodal planes of the  EMBED Equation.DSMT4  field are a distance  EMBED Equation.DSMT4  apart, so  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (b) By Eq.(32.36) we see that the nodal planes of the  EMBED Equation.DSMT4  field are also a distance  EMBED Equation.DSMT4  apart. (c)  EMBED Equation.DSMT4  Evaluate: The spacing between the nodes of  EMBED Equation.DSMT4  is the same as the spacing between the nodes of  EMBED Equation.DSMT4  Note that  EMBED Equation.DSMT4  as it must. 32.32. Identify: The nodal planes of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are located by Eqs.(32.26) and (32.27). Set Up:  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4  (b) The distance between the electric and magnetic nodal planes is one-quarter of a wavelength, so is  EMBED Equation.DSMT4  Evaluate: The nodal planes of  EMBED Equation.DSMT4  are separated by a distance  EMBED Equation.DSMT4  and are midway between the nodal planes of  EMBED Equation.DSMT4  32.33. (a) Identify and Set Up: The distance between adjacent nodal planes of  EMBED Equation.DSMT4  There is an antinodal plane of  EMBED Equation.DSMT4  midway between any two adjacent nodal planes, so the distance between a nodal plane and an adjacent antinodal plane is  EMBED Equation.DSMT4  Use  EMBED Equation.DSMT4  to calculate  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (b) Identify and Set Up: The nodal planes of  EMBED Equation.DSMT4  are at x = 0,  EMBED Equation.DSMT4  so the antinodal planes of  EMBED Equation.DSMT4  are at  EMBED Equation.DSMT4  The nodal planes of  EMBED Equation.DSMT4  are at  EMBED Equation.DSMT4  so the antinodal planes of  EMBED Equation.DSMT4  are at  EMBED Equation.DSMT4  Execute: The distance between adjacent antinodal planes of  EMBED Equation.DSMT4  and antinodal planes of  EMBED Equation.DSMT4  is therefore  EMBED Equation.DSMT4  (c) From Eqs.(32.36) and (32.37) the distance between adjacent nodal planes of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4  Evaluate: The nodes of  EMBED Equation.DSMT4  coincide with the antinodes of  EMBED Equation.DSMT4  and conversely. The nodes of  EMBED Equation.DSMT4  and the nodes of  EMBED Equation.DSMT4  are equally spaced. 32.34. Identify: Evaluate the derivatives of the expressions for  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 that are given in Eqs.(32.34) and (32.35). Set Up:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Similarly:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  (b) EMBED Equation.DSMT4 . EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Similarly:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Evaluate: The standing waves are linear superpositions of two traveling waves of the same k and  EMBED Equation.DSMT4 . 32.35. Identify: The nodal and antinodal planes are each spaced one-half wavelength apart. Set Up:  EMBED Equation.DSMT4  wavelengths fit in the oven, so  EMBED Equation.DSMT4  and the frequency of these waves obeys the equation f( = c. Execute: (a) Since  EMBED Equation.DSMT4  we have L = (5/2)(12.2 cm) = 30.5 cm. (b) Solving for the frequency gives f = c/( = (3.00 ( 108 m/s)/(0.122 m) = 2.46 ( 109 Hz. (c) L = 35.5 cm in this case.  EMBED Equation.DSMT4  so ( = 2L/5 = 2(35.5 cm)/5 = 14.2 cm. f = c/( = (3.00 ( 108 m/s)/(0.142 m) = 2.11 ( 109 Hz Evaluate: Since microwaves have a reasonably large wavelength, microwave ovens can have a convenient size for household kitchens. Ovens using radiowaves would need to be far too large, while ovens using visible light would have to be microscopic. 32.36. Identify: Evaluate the partial derivatives of the expressions for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  Execute: Assume  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Eq. (32.12) is  EMBED Equation.DSMT4 . This gives  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4  Similarly for Eq.(32.14),  EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4  Evaluate: The  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 fields must oscillate in phase. 32.37. Identify and Set Up: Take partial derivatives of Eqs.(32.12) and (32.14), as specified in the problem. Execute: Eq.(32.12):  EMBED Equation.DSMT4  Taking  EMBED Equation.DSMT4  of both sides of this equation gives  EMBED Equation.DSMT4 . Eq.(32.14) says  EMBED Equation.DSMT4  Taking  EMBED Equation.DSMT4  of both sides of this equation gives  EMBED Equation.DSMT4 . But  EMBED Equation.DSMT4  (The order in which the partial derivatives are taken doesn't change the result.) So  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  as was to be shown. Evaluate: Both fields, electric and magnetic, satisfy the wave equation, Eq.(32.10). We have also shown that both fields propagate with the same speed  EMBED Equation.DSMT4  32.38. Identify: The average energy density in the electric field is  EMBED Equation.DSMT4 and the average energy density in the magnetic field is  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4 . Execute:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 , which equals  EMBED Equation.DSMT4 . Evaluate: Our result allows us to write  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . 32.39. Identify: The intensity of an electromagnetic wave depends on the amplitude of the electric and magnetic fields. Such a wave exerts a force because it carries energy. 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Execute: (a) I = Pav/A = (25,000 W)/[4(575 m)2] = 0.00602 W/m2 (b)  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  = 2.13 N/C. Bmax = Emax/c = (2.13 N/C)/(3.00 ( 108 m/s) = 7.10 (  EMBED Equation.DSMT4  T (c) F =PavA = (I/c)A = (0.00602 W/m2)(0.150 m)(0.400 m)/(3.00 ( 108 m/s) = 1.20 (  EMBED Equation.DSMT4  N Evaluate: The fields are very weak compared to ordinary laboratory fields, and the force is hardly worth worrying about! 32.40. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . For a totally absorbing surface the radiation pressure is  EMBED Equation.DSMT4  Set Up: The wave speed in air is  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  Evaluate: The intensity depends only on the amplitudes of the electric and magnetic fields and is independent of the wavelength of the light. 32.41. (a) Identify and Set Up: Calculate I and then use Eq.(32.29) to calculate  EMBED Equation.DSMT4  and Eq.(32.18) to calculate  EMBED Equation.DSMT4  Execute: The intensity is power per unit area:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate: The magnetic field amplitude is quite small. (b) Identify and Set Up: Eqs.(24.11) and (30.10) give the energy density in terms of the electric and magnetic field values at any time. For sinusoidal fields average over  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  to get the average energy densities. Execute: The energy density in the electric field is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and the average value of  EMBED Equation.DSMT4  The average energy density in the electric field then is  EMBED Equation.DSMT4  The energy density in the magnetic field is  EMBED Equation.DSMT4  The average value is  EMBED Equation.DSMT4  Evaluate: Our result agrees with the statement in Section 32.4 that the average energy density for the electric field is the same as the average energy density for the magnetic field. (c) Identify and Set Up: The total energy in this length of beam is the total energy density  EMBED Equation.DSMT4  times the volume of this part of the beam. Execute:  EMBED Equation.DSMT4  Evaluate: This quantity can also be calculated as the power output times the time it takes the light to travel L = 1.00 m:  EMBED Equation.DSMT4  which checks. 32.42. Identify: Use the gaussian surface specified in the hint. Set Up: The wave is in free space, so in Gauss s law for the electric field,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Gauss s law for the magnetic field says  EMBED Equation.DSMT4  Execute: Use a gaussian surface such that the front surface is ahead of the wave front (no electric or magnetic fields) and the back face is behind the wave front, as shown in Figure 32.42.  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4   EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Evaluate: The wave must be transverse, since there are no components of the electric or magnetic field in the direction of propagation.  INCLUDEPICTURE FIG32-042.tif \* MERGEFORMAT \* MERGEFORMAT  Figure 32.42 32.43. Identify:  EMBED Equation.DSMT4 . For an absorbing surface, the radiation pressure is  EMBED Equation.DSMT4  Set Up: Assume the electromagnetic waves are formed at the center of the sun, so at a distance r from the center of the sun  EMBED Equation.DSMT4  Execute: (a) At the sun s surface:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  (b) Halfway out from the sun s center, the intensity is 4 times more intense, and so is the radiation pressure:  EMBED Equation.DSMT4 and EMBED Equation.DSMT4  At the top of the earth s atmosphere, the measured sunlight intensity is  EMBED Equation.DSMT4 , which is about 100,000 times less than the values above. Evaluate: (b) The gas pressure at the sun s surface is 50,000 times greater than the radiation pressure, and halfway out of the sun the gas pressure is believed to be about  EMBED Equation.DSMT4  times greater than the radiation pressure. Therefore it is reasonable to ignore radiation pressure when modeling the sun s interior structure. 32.44. Identify:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4  Execute:  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Evaluate: This value of  EMBED Equation.DSMT4 is similar to the electric field amplitude in ordinary light sources. 32.45. Identify: The same intensity light falls on both reflectors, but the force on the reflecting surface will be twice as great as the force on the absorbing surface. Therefore there will be a net torque about the rotation axis. Set Up: For a totally absorbing surface,  EMBED Equation.DSMT4  while for a totally reflecting surface the force will be twice as great. The intensity of the wave is  EMBED Equation.DSMT4  Once we have the torque, we can use the rotational form of Newton s second law, (net = I(, to find the angular acceleration. Execute: The force on the absorbing reflector is  EMBED Equation.DSMT4  For a totally reflecting surface, the force will be twice as great, which is  EMBED Equation.DSMT4 . The net torque is therefore  EMBED Equation.DSMT4  Newton s 2nd law for rotation gives  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Solving for ( gives  EMBED Equation.DSMT4  Evaluate: This is an extremely small angular acceleration. To achieve a larger value, we would have to greatly increase the intensity of the light wave or decrease the mass of the reflectors. 32.46. Identify: For light of intensity  EMBED Equation.DSMT4 incident on a totally absorbing surface, the radiation pressure is  EMBED Equation.DSMT4 . For light of intensity EMBED Equation.DSMT4  incident on a totally reflecting surface,  EMBED Equation.DSMT4 . Set Up: The total radiation pressure is  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Execute: (a)  EMBED Equation.DSMT4 . (b) (i) For totally absorbing  EMBED Equation.DSMT4 . (ii) For totally reflecting  EMBED Equation.DSMT4  These are just equations 32.32 and 32.33. (c) For  EMBED Equation.DSMT4   EMBED Equation.DSMT4  For  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate: The radiation pressure is greater when a larger fraction is reflected. 32.47. Identify and Set Up: In the wire the electric field is related to the current density by Eq.(25.7). Use Ampere s law to calculate  EMBED Equation.DSMT4  The Poynting vector is given by Eq.(32.28) and the equation that follows it relates the energy flow through a surface to  EMBED Equation.DSMT4  Execute: (a) The direction of  EMBED Equation.DSMT4  is parallel to the axis of the cylinder, in the direction of the current. From Eq.(25.7),  EMBED Equation.DSMT4  (E is uniform across the cross section of the conductor.) (b) A cross-sectional view of the conductor is given in Figure 32.47a; take the current to be coming out of the page.  INCLUDEPICTURE FIG32-047a.tif \* MERGEFORMAT \* MERGEFORMAT Apply Ampere s law to a circle of radius a.  EMBED Equation.DSMT4   EMBED Equation.DSMT4 Figure 32.47a EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  The direction of  EMBED Equation.DSMT4  is counterclockwise around the circle. (c) The directions of  EMBED Equation.DSMT4  are shown in Figure 32.47b.  INCLUDEPICTURE FIG32-047b.tif \* MERGEFORMAT \* MERGEFORMAT The direction of  EMBED Equation.DSMT4  is radially inward.  EMBED Equation.DSMT4   EMBED Equation.DSMT4 Figure 32.47b(d) Evaluate: Since S is constant over the surface of the conductor, the rate of energy flow P is given by S times the surface of a length l of the conductor:  EMBED Equation.DSMT4  But  EMBED Equation.DSMT4  so the result from the Poynting vector is  EMBED Equation.DSMT4  This agrees with  EMBED Equation.DSMT4  the rate at which electrical energy is being dissipated by the resistance of the wire. Since  EMBED Equation.DSMT4  is radially inward at the surface of the wire and has magnitude equal to the rate at which electrical energy is being dissipated in the wire, this energy can be thought of as entering through the cylindrical sides of the conductor. 32.48. Identify: The intensity of the wave, not the electric field strength, obeys an inverse-square distance law. Set Up: The intensity is inversely proportional to the distance from the source, and it depends on the amplitude of the electric field by I = Sav =  EMBED Equation.DSMT4 cEmax2. Execute: Since I =  EMBED Equation.DSMT4 cEmax2,  EMBED Equation.DSMT4 . A point at 20.0 cm (0.200 m) from the source is 50 times closer to the source than a point that is 10.0 m from it. Since I ( 1/r2 and (0.200 m)/(10.0 m) = 1/50, we have I0.20 = 502 I10. Since  EMBED Equation.DSMT4 , we have E0.20 = 50E10 = (50)(1.50 N/C) = 75.0 N/C. Evaluate: While the intensity increases by a factor of 502 = 2500, the amplitude of the wave only increases by a factor of 50. Recall that the intensity of any wave is proportional to the square of its amplitude. 32.49. Identify and Set Up: The magnitude of the induced emf is given by Faraday s law:  EMBED Equation.DSMT4  To calculate  EMBED Equation.DSMT4  we need  EMBED Equation.DSMT4  at the antenna. Use the total power output to calculate I and then combine Eq.(32.29) and (32.18) to calculate  EMBED Equation.DSMT4  The time dependence of B is given by Eq.(32.17). Execute:  EMBED Equation.DSMT4  where R = 0.0900 m is the radius of the loop. (This assumes that the magnetic field is uniform across the loop, an excellent approximation.)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The maximum value of  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Calculate the intensity I at this distance from the source, and from that the magnetic field amplitude  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Thus  EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  Evaluate: An induced emf of this magnitude is easily detected. 32.50. Identify: The nodal planes are one-half wavelength apart. Set Up: The nodal planes of B are at x = (/4, 3(/4, 5(/4, & , which are (/2 apart. Execute: (a) The wavelength is ( = c/f = (3.000 ( 108 m/s)/(110.0 ( 106 Hz) = 2.727 m. So the nodal planes are at (2.727 m)/2 = 1.364 m apart. (b) For the nodal planes of E, we have (n = 2L/n, so L = n(/2 = (8)(2.727 m)/2 = 10.91 m Evaluate: Because radiowaves have long wavelengths, the distances involved are easily measurable using ordinary metersticks. 32.51. Identify and Set Up: Find the force on you due to the momentum carried off by the light. Express this force in terms of the radiated power of the flashlight. Use this force to calculate your acceleration and use a constant acceleration equation to find the time. (a) Execute:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Then  EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4  Evaluate: The radiation force is very small. In the calculation we have ignored any other forces on you. (b) You could throw the flashlight in the direction away from the ship. By conservation of linear momentum you would move toward the ship with the same magnitude of momentum as you gave the flashlight. 32.52. Identify:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Set Up: The power carried by the current i is  EMBED Equation.DSMT4 . Execute:  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate:  EMBED Equation.DSMT4 . This is a very intense beam spread over a large area. 32.53. Identify: The orbiting satellite obeys Newton s second law of motion. The intensity of the electromagnetic waves it transmits obeys the inverse-square distance law, and the intensity of the waves depends on the amplitude of the electric and magnetic fields. Set Up: Newton s second law applied to the satellite gives mv2/R = GmM/r2, where M is the mass of the Earth and m is the mass of the satellite. The intensity I of the wave is I = Sav =  EMBED Equation.DSMT4 cEmax2, and by definition, I = Pav/A. Execute: (a) The period of the orbit is 12 hr. Applying Newton s 2nd law to the satellite gives mv2/R = GmM/r2, which gives  EMBED Equation.DSMT4 . Solving for r, we get  EMBED Equation.DSMT4  The height above the surface is h = 2.66 ( 107 m  6.38 ( 106 m = 2.02 ( 107 m. The satellite only radiates its energy to the lower hemisphere, so the area is 1/2 that of a sphere. Thus, from the definition of intensity, the intensity at the ground is I = Pav/A = Pav/(2h2) = (25.0 W)/[2(2.02 ( 107 m)2] = 9.75 (  EMBED Equation.DSMT4  W/m2 (b) I = Sav =  EMBED Equation.DSMT4 cEmax2, so  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (c) Pav = I/c = (9.75(10 15 W/m2)/(3.00(108 m/s) = 3.25(10 23 Pa (d) ( = c/f = (3.00(108 m/s)/(1575.42(106 Hz) = 0.190 m Evaluate: The fields and pressures due to these waves are very small compared to typical laboratory quantities. 32.54. Identify: For a totally reflective surface the radiation pressure is  EMBED Equation.DSMT4  Find the force due to this pressure and express the force in terms of the power output P of the sun. The gravitational force of the sun is  EMBED Equation.DSMT4  Set Up: The mass of the sum is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Execute: (a) The sail should be reflective, to produce the maximum radiation pressure. (b)  EMBED Equation.DSMT4 , where A is the area of the sail.  EMBED Equation.DSMT4 , where r is the distance of the sail from the sun.  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  so  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . (c) Both the gravitational force and the radiation pressure are inversely proportional to the square of the distance from the sun, so this distance divides out when we set  EMBED Equation.DSMT4  Evaluate: A very large sail is needed, just to overcome the gravitational pull of the sun. 32.55. Identify and Set Up: The gravitational force is given by Eq.(12.2). Express the mass of the particle in terms of its density and volume. The radiation pressure is given by Eq.(32.32); relate the power output L of the sun to the intensity at a distance r. The radiation force is the pressure times the cross sectional area of the particle. Execute: (a) The gravitational force is  EMBED Equation.DSMT4  The mass of the dust particle is  EMBED Equation.DSMT4  Thus  EMBED Equation.DSMT4  (b) For a totally absorbing surface  EMBED Equation.DSMT4  If L is the power output of the sun, the intensity of the solar radiation a distance r from the sun is  EMBED Equation.DSMT4  Thus  EMBED Equation.DSMT4  The force  EMBED Equation.DSMT4  that corresponds to  EMBED Equation.DSMT4  is in the direction of propagation of the radiation, so  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the component of area of the particle perpendicular to the radiation direction. Thus  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Evaluate: The gravitational force and the radiation force both have a  EMBED Equation.DSMT4  dependence on the distance from the sun, so this distance divides out in the calculation of R. (d)  EMBED Equation.DSMT4  is proportional to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is proportional to  EMBED Equation.DSMT4  so this ratio is proportional to 1/R. If  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4  and the radiation force will drive the particles out of the solar system. 32.56. Identify: The electron has acceleration  EMBED Equation.DSMT4 . Set Up:  EMBED Equation.DSMT4 . An electron has  EMBED Equation.DSMT4 . Execute: For the electron in the classical hydrogen atom, its acceleration is  EMBED Equation.DSMT4  Then using the formula for the rate of energy emission given in Problem 32.57:  EMBED Equation.DSMT4  This large value of  EMBED Equation.DSMT4  would mean that the electron would almost immediately lose all its energy! Evaluate: The classical physics result in Problem 32.57 must not apply to electrons in atoms. 32.57. Identify: The orbiting particle has acceleration  EMBED Equation.DSMT4  Set Up:  EMBED Equation.DSMT4 . An electron has mass  EMBED Equation.DSMT4  and a proton has mass  EMBED Equation.DSMT4 . Execute: (a)  EMBED Equation.DSMT4  (b) For a proton moving in a circle, the acceleration is  EMBED Equation.DSMT4 The rate at which it emits energy because of its acceleration is  EMBED Equation.DSMT4  Therefore, the fraction of its energy that it radiates every second is  EMBED Equation.DSMT4  (c) Carry out the same calculations as in part (b), but now for an electron at the same speed and radius. That means the electron s acceleration is the same as the proton, and thus so is the rate at which it emits energy, since they also have the same charge. However, the electron s initial energy differs from the proton s by the ratio of their masses:  EMBED Equation.DSMT4  Therefore, the fraction of its energy that it radiates every second is  EMBED Equation.DSMT4  Evaluate: The proton has speed  EMBED Equation.DSMT4 . The electron has the same speed and kinetic energy 3.27 keV. The particles in the accelerator radiate at a much smaller rate than the electron in Problem 32.56 does, because in the accelerator the orbit radius is very much larger than in the atom, so the acceleration is much less. 32.58. Identify and Set Up: Follow the steps specified in the problem. Execute: (a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Setting  EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4 . This will only be true if  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4  (b) The energy in the wave is dissipated by the  EMBED Equation.DSMT4  heating of the conductor. (c)  EMBED Equation.DSMT4  Evaluate: The lower the frequency of the waves, the greater is the distance they can penetrate into a conductor. 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FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfo 8Equation Native 9_1240637167  FMPMPOle IO xDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A   dEdt== q 2 a 2 6pWinANSIWWdoc08PP 0 c 3 == (1.610 "-19  C) 2 (1.5310 15   ms 2 ) 2 6pPP 0 (3.010 8   ms) 3 ==1.3310 "-23   Js==8.3210 "-5   eVs. FMathType 5.0 Equation MathType EFEquation.DSMT49q9$DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_ECompObj JiObjInfo LEquation Native M#_1240637170 FMPMP_A   ( dEdt)(1s)E== 8.3210 "-5 eV6.0010 6 eV==1.3910 "-11 . FMathType 5.0 Equation MathTyOle VCompObj WiObjInfo YEquation Native Zpe EFEquation.DSMT49q9vLDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A  E e ==E p  m e m p ==(6.0010 6 eV) (9.1110 "-31 kg)(1.6710 "-27 kg)==3273eV. FMathType 5.0 Equation MathType EFEquation.DSMT49q_1240637172 FMPPOle eCompObj fiObjInfo h9<DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A    (dEdt)(1s)E== 8.3210 "-5 eV3273eV==2Equation Native i_1240637174 FPPOle qCompObj ri.5410 "-8 . FMathType 5.0 Equation MathType EFEquation.DSMT49q9zDDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_EObjInfo tEquation Native u_1240637177 FPPOle _A  v==  2Em p ==  2(6.010 6  eV)(1.6010 "-19  J/eV)1.6710 "-27  kg  ==3.3910 7  m/s FMathType 5.0 Equation MathType EFEquation.DSMT49q9DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A  E y (xCompObj iObjInfo Equation Native _1240637179 FPP,t)==E max e "-k C x sin(k C x"-wt). FMathType 5.0 Equation MathType EFEquation.DSMT49qOle CompObj iObjInfo Equation Native 94DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A   "E y "x==E max ("-k C )e "-k C x sin(k C x"-wt)++E max (++k C )e "-k C x cos(k C x"-wt) FMathType 5.0 Equation MathType EFEquation.DSMT49qF¹ xDSMT5WinAllBasicCodePages_1242713449 FPPOle CompObj iObjInfo Equation Native _1240637184 FPPOle CompObj iTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A   " 2 E y "x 2 ==E max (++k C2 )e "-k C x sin(k C x"-wt)++E max ("-k C2 )e "-k C x cos(k C x"-wt) FMathType 5.0 Equation MathType EFEquation.DSMT49q9pLDSMT5WinAllBasicCodePagesObjInfo Equation Native _1240637186 FPPOle Times New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A  ++E max ("-k C2 )e "-k C x cos(k C x"-wt)++E max ("-k C2 )e "-k C x sin(k C x"-wt) FMathType 5.0 Equation MathType EFEquation.DSMT49q9<DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBACompObj  iObjInfo Equation Native _1240637188  FPPu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A   " 2 E y "x 2 =="-2E max k C2 e "-k C x cos(k C x"-wt)Ole CompObj  iObjInfo Equation Native  FMathType 5.0 Equation MathType EFEquation.DSMT49q9DDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra)XBAu!APAE%B_AC_AE*_HA@AHA*_D_E_E_A   "E y "t==E max e "-k C x wcos(k C x"-wt). 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