A beam of diameter 'd' is incident on a glass hemisphere as shown. If the radius of curvature of the hemisphere is very large in comparison to d, then the diameter of the beam at the base of the hemisphere will be :

Options

A. ` ( 3 ) /( 4 ) ` d
B. d
C. ` ( d ) /( 3 ) `
D. ` ( 2 ) / ( 3 ) d `

Answer

D

Explanation

Applying the formula for spherical curved surface :
` ( mu _ 2 ) /( v ) - ( mu _ 1 ) /( u ) = ( mu_ 2 - mu _ 1 ) /( R ) rArr ( 3 ) /( 2v ) - ( 1 ) /( oo) = ((3 ) /(2 ) - 1 )/ ( R) `
` rArr v = 3R `
Let x be the diameter of the light falling at base.
` (d ) / ( x ) = ( 3R) / ( 2R ) ` ( similar triangle concept ) ` rArr x = ( 2 ) /( 3 ) d ` .