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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 ¼ p LC , where C is the capacitance and L is the inductance. Thus ¸ 2 ¼ p LC = c: The solution for L is L = ¸ 2 4 ¼ 2 Cc 2 = (550 £ 10 ¡ 9 m) 2 4 ¼ 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 crosssectional 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 ¼f = 2 ¼ (1 : £ 10 8 Hz) = 6 : 3 £ 10 8 rad = s : (c) The angular wave number is k = 2 ¼ ¸ = 2 ¼ 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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 Fall '98
 Heckman
 Physics, Current

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