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Unformatted text preview: Chapter 33 5 If f is the frequency and is the wavelength of an electromagnetic wave, then f = c . The frequency is the same as the frequency of oscillation of the current in the LC circuit of the generator. That is, f = 1 = 2 p LC , where C is the capacitance and L is the inductance. Thus 2 p LC = c: The solution for L is L = 2 4 2 Cc 2 = (550 10 9 m) 2 4 2 (17 10 12 F)(3 : 00 10 8 m = s) 2 = 5 : 00 10 21 H : This is exceedingly small. 21 The plasma completely reflects all the energy incident on it, so the radiation pressure is given by p r = 2 I=c , where I is the intensity. The intensity is I = P=A , where P is the power and A is the area intercepted by the radiation. Thus p r = 2 P Ac = 2(1 : 5 10 9 W) (1 : 00 10 6 m 2 )(3 : 00 10 8 m = s) = 1 : 10 7 Pa = 10 MPa : 23 Let f be the fraction of the incident beam intensity that is reflected. The fraction absorbed is 1 f . The reflected portion exerts a radiation pressure of p r = (2 fI ) =c and the absorbed portion exerts a radiation pressure of p a = (1 f ) I =c , where I is the incident intensity. The factor 2 enters the first expression because the momentum of the reflected portion is reversed. The total radiation pressure is the sum of the two contributions: p total = p r + p a = 2 fI + (1 f ) I c = (1 + f ) I c : To relate the intensity and energy density, consider a tube with length ` and cross-sectional area A , lying with its axis along the propagation direction of an electromagnetic wave. The electromagnetic energy inside is U = uA` , where u is the energy density. All this energy will pass through the end in time t = `=c so the intensity is I = U At = uA`c A` = uc: Thus u = I=c . The intensity and energy density are inherently positive, regardless of the propa- gation direction. 208 Chapter 33 For the partially reflected and partially absorbed wave, the intensity just outside the surface is I = I + fI = (1 + f ) I , where the first term is associated with the incident beam and the second is associated with the reflected beam. The energy density is, therefore, u = I c = (1 + f ) I c ; the same as radiation pressure. 25 (a) Since c = f , where is the wavelength and f is the frequency of the wave, f = c = 3 : 00 10 8 m = s 3 : 0 m = 1 : 10 8 Hz : (b) The angular frequency is ! = 2 f = 2 (1 : 10 8 Hz) = 6 : 3 10 8 rad = s : (c) The angular wave number is k = 2 = 2 3 : 0 m = 2 : 1 rad = m : (d) The magnetic field amplitude is B m = E m c = 300 V = m 3 : 00 10 8 m = s = 1 : 00 10 6 T : (e) ~ B must be in the positive z direction when ~ E is in the positive y direction in order for ~ E ~ B to be in the positive x direction (the direction of propagation)....
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