Answer
distance traveled in vacuum is $4.316cm$
Work Step by Step
Given that distance travelled by light in ice $d_{ice}=2.0cm=2.0\times10^{-2}m$
distance travelled by light in crystaline quartz $d_{qz}=1.1cm=1.1\times10^{-2}m$
From table 26.1
Refractive index of ice $n_{ice}=1.309$
Refractive index of crystalline quartz is $n_{qz}=1.544$
refractive index is defined as
$n=\frac{c}{v}$
$c$=speed of light in vacuum
$v$= speed of light in material
we can rewrite above equation as
$v=\frac{c}{n}$
as speed of light in vacuum $c=3\times10^8m/s$
speed of light in ice
$v_{ice}=\frac{c}{n_{ice}}=\frac{3\times10^8m/s}{1.309}=2.2918\times^{8}m/s$
suppose time taken by the light to pass through ice is $t_{ice}$
from $time=\frac{distance}{speed}$
$t_{ice}=\frac{d_{ice}}{v_{ice}}=\frac{2.0\times10^{-2}m}{2.2918\times^{8}m/s}$=$0.87267\times^{-10}s$........equation(1)
speed of light in quartz
$v_{qz}=\frac{c}{n_{qz}}=\frac{3\times10^8m/s}{1.544}=1.943\times^{8}m/s$
suppose time taken by the light to pass through ice is $t_{qz}$
from $time=\frac{distance}{speed}$
$t_{qz}=\frac{d_{qz}}{v_{qz}}=\frac{1.1\times10^{-2}m}{1.9430\times^{8}m/s}$=$0.56614\times^{-10}s$........equation(2)
from equation (1) & (2)total time taken by light to pass through ice and quartz will be
$t=t_{ice}+t_{qz}$
$t=0.87267\times^{-10}s+0.56614\times^{-10}s=1.43881\times^{-10}s$
speed of light in vacuum $c=3\times10^8m/s$
so in vacuum distance travelled in time $t$ will be
$distance=speed\times time=c\times t=3\times10^8m/s\times1.43881\times^{-10}s$
distance traveled in vacuum is $4.316 \times^{-2}m$=$4.316cm$
