In the figure shown , for an angle of incidence `45^(@)`, at the top surface , what is the minimum refractive index needed for the internal reflection at vertical face ?

Options

A. `(sqrt2+1)/2`
B. `sqrt(3/2)`
C. `sqrt(1/2)`
D. `sqrt(2)`

Answer

B

Explanation

(b) At point P,

Refractive index,`mu=(sin45^(@))/(sinr)rArrsinr=1/(musqrt2)`……(i)
At point Q, for total internal reflection,
`sini'=1/mu`
From figure, I'=`90^(@)-r`
`therefore sin(90^(@)-r)=1/murArrcos r=1/mu`....(ii)
Now, `cosr=sqrt(1-sin^(2)r)=sqrt((1-1/(2mu^(2))))=sqrt(((2mu^(2)-1)/(2mu^(2))))`.....(iii)
from Eqs. (ii) and (iii)
`1/mu=sqrt((2mu^(2)-1)/(2mu^(2)))`
`rArr` Refractive index, `mu=sqrt(3/2)`
For total internal reflection at verticle face AD.