(a) Fig. shows a cross-section of a 'light pipe' made of a glass fibre of refractive index `1.68`. The outer covering of the pipe is made of a material of refractive index `1.44`. What is the axis of the pipe for which total reflection inside the pipe take place as shwon.
(b) What is the answer if there is no outer covering if the pipe ?

Explanation

(a) Given, refractive index of glass fibre, `mu_(1)=1.68`
Refractive index of outer covering, `mu_(2)=1.44`
Let `i_(c )` be the critical angle for this interface.
`sin i_(c )=(mu_(2))/(mu_(1))=(1.44)/(1.68)=0.8571`
`therefore" "i_(c )=59^(@)`
Total internal reflection takes place when `i. gt 59^(@)` or angle r may have value less than `r_("max")` which is given by
`therefore" "r_("max")=9-^(@)-i_(c )=90^(@)-59^(@)=31^(@)`
`"Now "(sin i_("max"))/(sinr_("max"))=1.68`
`(sini_("max"))/(sin 31^(@))=1.68`
`sini_("max")=0.8652`
`therefore" "i_("max")~~60^(@)`
Thus, all incident rays of angles in the range `0lt i lt 60^(@)` will suffer total internal reflection in the pipe.
(b) If there is no outer coating of the pipe,
`sini._(c )=(1)/(mu_(1))`
`sini._(c)=(1)/(1.68)=0.5952`
`"or "sini._(c)=sin36.5^(@)`
`"or "i._(c )=36.5^(@)`
If `i=90^(@)` then `r=36.5^(@)` and `i.=90^(@)-r=90^(@)-36.5^(@)=53.5^(@)` which is greater than `i._(c )`. Thus all incident rays (in the range `0ltilt90^(@)`) will suffer total internal reflection.