A uniform, horizontal beam of light is incident upon a quarter cyclinder of radius R=5cm, and has a refractive index` 2//sqrt(3)`. A patch on the table at a distance 'x' from the cylinder is unilluminated. Find the value of 'x'?

Explanation

When the angle of incidence is , .x. region will remain unillumiated. This is the limiting case.

When the angle of icidence is greater than critical agle T.I.R. will place, In this case also .x. region is unillaminated.

When angle of incidence `theta lt theta_c` , then the incident ray will refract and hence .x. region will be illuminated. So, in limiting case:

In ` Delta PQR , cos theta theta_c = (R )/(R +x) `
` sin theta_c = (mu_r )/(mu _d )=(1)/( 2 //sqrt(3)) = ( sqrt(3))/(2)`
` therefore theta_c = 60^@ , R = 5 cm `( given )
`implies cos 60 = (5)/( 5+x) implies 5 +x =10`
` implies x=5 cm`.