Refraction of Electric Waves. 549 



must substitute this value of f in (2) . Writing V = . — -, we 

 obtain V */* 



y 2 __ %tt 2 _ , /4tt4 16tt 4 CV 

 * - T2 y o2 ty ^ + — ^ , 



2 _ J _ 2tt* /^ 



16tt 4 0V 



The solution (3) is imaginary, but two such solutions 

 properly combined give, as a real solution of (2), 



/=A«*.cos(*p+^). 



The factor e& gives the amplitude of the wave ; and 



~ = V, the velocity of propagation. 

 77I 



Whence , 



v=1 /\/ 2 4? + x/^+cVT 2 ; 



When 2 is negligible in comparison with C (i. e. when 

 the conductivity is small), this expression expanded gives 



V = V - V ^^; (4) 



and similarly, 



f=-ftrO/*V, (5) 



The units here used are the electromagnetic system ; hence 

 V , from its definition — jr=. , is the velocity with which the 



wave would move in the given medium if its conductivity were 

 zero. 



Now let D =the maximum amplitude of a wave on entering 

 a sheet of the dielectric of thickness z ; 

 D = maximum amplitude of wave on emerging. 



Then 



D 



= D ^*°. 



Call 



D 

 Do 



=?; 



Then 





log q = %; =—27rCfiV z ; 



whence 





p 2 ^_ lo z A q 



