Surfaces of a thin equi-convex glass lens have radius of curvature R. Paraxial rays are incident on it . If the final image is formed after n internal reflections, calculate distance of this image from pole of the lens. Refractive index of glass is `mu`.

Explanation

The rays will first get refracted, then n-times reflected and finally again refracted.
So, using `mu_2/v-mu_1/u=(mu_2-mu_1)/R` for first refraction, we have

`mu/v_i-1/oo=mu-1/R rArr v_i=((mu)/(mu-1))R`
For first reflection, let us use `1/v+1/u=1/f=2/R`
`:. 1/v_1+((mu-1)/(muR))=-2/R or 1/v_1=-((3mu-1)/(muR))`
For second reflection, `1/v_2+((3mu-1)/(muR))=-2/R` or `1/v_2=-((5mu-1)/(muR))`
Similarly, after `n^(th)` reflection, `1/v_n=-[((2n+1)mu-1)/(muR)]`
Finally, using `mu_2/v-mu_1/u=(mu_2-mu_1)/R`, we have
`1/v_f=-{((2n+1)mu-1)/R}=(1-mu)/-R`
`v_f=R/(2(mun+mu-1))`