520 The Electromagnetic Theory of Light [OH. xvm 



1 iv 



Since v is real, we have 



1 + iv 



= 1, so that we may take 



where % = arg ~ = - 2 tan- 1 *. 



In the reflected wave, we now have 



y _ ym i*\ ( - x cos 0, +y sin 0j - F! 

 _ y iit l (-xcosOi+ysin0 1 Vi 



- JLJ 6 



Comparing with the incident wave, in which 



17 r?' fceiteeostfi+yrin^-FiO 

 Z = zj e , 



we see that reflection is now accompanied by a change of phase - 2/e tan -1 t;, 

 but the amplitude of the wave remains unaltered, as obviously it must from 

 the principle of energy. 



Refraction of a Wave polarised perpendicular to plane of incidence. 



597. The analysis which has been already given can easily be modified 

 so as to apply to the case in which the polarisation of the incident wave is 

 perpendicular to the plane of incidence. All that is necessary is to inter- 

 change corresponding electric and magnetic quantities : we then have an 

 incident wave in which the magnetic force is perpendicular to the plane of 

 incidence, and this is what is required. 



Clearly all the geometrical laws which have already been obtained will 

 remain true without modification, and the analyses of 591 (total reflection) 

 will also hold without modification. 



Formula (563), giving the amplitude of the reflected ray, will, however, 

 require alteration. We still have 



but the value of u, instead of being given by equation (563), must now be 

 supposed to be given by 



^ K, cos 2 #2 



u T^ ~ rz~ > 

 K 2 /xj cos 2 ^ 



this equation being obtained by the interchange of electric and magnetic 

 terms in equation (562). Taking yu 2 = /^ = 1, we obtain 



*i cos 2 sin 6 2 cos $2 si n 2$ 2 

 T 2 cos ^ sin 9 l cos ^ sin 2^ x ' 



whence, from equation (565), -^ ^ 2 ~ (566), 



^ tan (#2 -f- ^) 



giving the ratio of the amplitudes of the incident and reflected waves. This 

 result also agrees well with experiment. 



