A ray of light strikes a glass slab of thickness t.
(i) Prove that it emerges on the opposite face, parallel to the initial ray.
(ii) (ii) Prove that the value of deflection of beam which passed through the plate is :
`t sini_(1)[1-sqrt(((1-sin^(2)i_(1))/(._(a)mu_(g)^(2)-sin^(1)i_(1))))]`
(iii) Prove that for a small angle of incidence `i_(1)`, the internal shift x is given by
`x=t i_(1)(1-(1)/(._(a)mu_(g)))`
where `._(a)mu_(g)` is the refractive index of glass with respect to air.

Explanation

Lateral shift, `y=BC =AB sinr^(.)=AB sin(theta-r)`
`implies y= t/(cosr) (sintheta - costhetasinr) = t(sintheta - costhetatanr)`
`tsintheta(1- cot thetatanr)`…(2)
where AB= t/cosr and r= angle of refraction given by using snell.s law,
`(sintheta)/(sinr) =n`
`implies sinr = (sintheta)/n` & cosr in (1) we obtain
`y=tsintheta[1- cottheta ((sintheta)/n)/(sqrt(n^(2)-sin^(2)theta)/n)]`
`=tsintheta[1- sqrt((1-sin^(2)theta)/(n^(2)-sin^(2)theta))]`
(b) If `theta` is small, `sin^(2)theta` can be ignored in comparison to 1, `n^(2)`
`implies y cong t theta[1- sqrt(1/n^(2))]`
`implies y = t theta[1-1/n]`