A cylindrical glass rod of radius 0.1m and refractive index `sqrt(3)` lies on horizontal plane mirror. A horizontal ray of light moving perpendicular to the axis of the rod is incident on it. At what height from the mirror should the ray be incident so that it leaves the rod at a height of 0.1m above the plane mirror ? At what distance a second similar rod, parallel to the first, be placed on the mirror, such that the emergent ray from the second rod is in line with the incident ray on the first rod ?

Explanation

The ray diagram as described in question is as follows :

(a) From the figure we can write the following : `i=2r`
Further we can use this in Snell.s law as follows :
`mu sin theta =` constant
`implies " " 1 xx sin i= mu xx sin r`
`implies sin i= mu sin r`
`implies sin 2r = mu sin r`
`implies sin 2r = mu sin r`
`implies 2 sin r cos r = mu sin r`
`implies cos r = (mu)/(2) " " ... (i)`
`implies sin r = sqrt(1 - (mu^(2))/(4)) " " .... (ii)`
Using figure we can write the following :
sin i = `(h)/(R)`
`implies h = R sin i`
`implies h = R sin 2r`
`implies h = 2 R sin r cos r `
Substituting from equation (i) and (ii) we get the following :
`implies h = 2 R xx sqrt(1 - (mu^2)/(4)) xx (mu)/(2)`
`implies h = (muR)/(2) sqrt(4 - mu^(2))`
(b) Emergent ray from the first sphere strikes the mirror at point T . And after the point T other sphere is to be placed symmetrically as shown in the figure . Hence , distance between centres of the two spheres will be `2 (R + a)`. Hence , to calculate the distance of the other sphere we need to first find a , as shown in figure .
From the figure we can write the following :
a = R cot i
`implies a = R cot 2r`
Distance between the centres of spheres can be written as follows :
`d = 2 (R + a)`
`implies d = 2 (R + R cot 2r)`
`implies d = 2 R( 1 + cot r)`
`implies d = 2 R(1 + (cos 2r)/(sin 2r))`
`implies d = 2 R (1 + (2 cos^(2) r - 1)/(2 sin r cos r ))`
`implies d = 2 R ( 1 + (2 ((mu^(2))/(4)) - 1)/(2 xx sqrt(1 - (mu^(2))/(4)) xx (mu)/(2)))`
`implies d = 2 R (1 + (mu^(2) - 2)/(mu //sqrt(4 - mu^(2))))`