Question

Shown in Figure P25.55 is a shallow glass dish whose width w is 4.20 cm. When an observer's eye is placed as shown, the observer sees the edge of the bottom of the empty dish. When this dish is filled with glycerin, the observer sees the center of the bottom of the dish. Find the height h of the dish.

Answer 1

Here are the simple diagram which I have made from the question description

so,

tan = w / H

when it is filled with glycerin

n1sin = n2sin'

where

tan ' = w/2H

put both tan equations in snell's law equation

n1 * sqrt ( (w/2H)2 / 1 + (w/2H)2 ) = n2 * sqrt ( (w/H)2 / 1 + (w/H)2 )

simplify, we get (squaring and cross multiplying)

n12 * (w/2H)2 * (1 + (w/H)2 ) = n22 * (w/H)2 * ( 1 + (w/2H)2 )

H = w * sqrt ( ( n12 - n22 ) / ( 4n22 - n12))

where n2 = 1 (air) and n1 = 1.473 ( glycerin)

so,

H = 4.20 * sqrt ( ( 1.4732 - 12 ) / ( 4*12 -1.4732))

H = 4.20 * sqrt ( 1.1697 / 1.83027)

H = 3.3575 cm

(PLEASE LET ME KNOW IF YOU HAVE ANY DOUBT)