594-596] Non-conducting Media 519 



For all media in which light can be propagated, we may take p = 1, so 

 that 



cos 0. 2 sin #! cos 2 tan 6 l 



IK, 



~V 7T 



___________ _ ............ (563). 



-L cos 6 1 sin # 2 cos 0^ tan 2 



Thus the ratio of the amplitude of the reflected to the incident ray is 

 Z'" _ 1 u tan #2 tan 6 l sin (0 2 #1) 



~ 1 + u ~ tan 2 + tan 0, sin (ft + 0,)" ' 

 This prediction of the theory is in good agreement with experiment. 



Zt" 



This being so, the predicted ratio of -^ is necessarily in agreement with 



^ 



experiment, since both in theory and experiment the energy of the incident 

 wave must be equal to the sum of the energies of the reflected and refracted 

 waves. 



Total Reflection. 

 596. We have seen (equation (557)) that the angle 2 is given by 



sin 2 = - sin O l , 

 n 



where n is the index of refraction for light passing from medium 1 to 

 medium 2. If n is less than unity, the value of - sin O l may be either 



greater or less than unity according as ^ > or < sin" 1 n. In the former 

 case sin 2 is greater than unity, so that the value of # 2 is imaginary. 



This circumstance does not affect the value of the foregoing analysis in a 

 case in which 6 > sin" 1 n, but the geometrical interpretation no longer holds. 



Let us denote - sin 0j by p, and Vj 2 I by q. Then in the analysis we 



n 



may replace sin # 2 by p, and cos # 2 by iq, both p and q being real quantities. 

 The exponential which occurs in the refracted wave is now 



iK 2 (x cos 9 2 +y sin 2 V^t) 



Thus the refracted wave is propagated parallel to the axis of y, i.e. 

 normal to the boundary, and its magnitude decreases proportionally to the 

 factor e~ K * qx . At a small distance from the boundary the refracted wave 

 becomes imperceptible. 



Algebraically, the values of Z' t Z" and Z'" are still given by equations (5b'l), 

 but we now have 



u / K ^ cos ^ = j / K ^ 

 v ft 2 ATi cos #! V jjb z K l 



l cos #j ' 



so that u is an imaginary quantity, say u = iv, and, from equations (561), 



Z^_ ^ \-u = l-iv 

 Z' ~~ 



