Question

lateral displacement take place due to refraction in a

Answer 1

Lateral displacement take place due to refraction in a Glass slab
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Answer 2

lateral displacement is the perpendicular distance between the path of emergent Ray and the direction of incident ray. it depends on the thickness of the block of the medium , the angle of incidence and the refractive index of Glass and therefore also on the wavelength of light used.
Suppose your slab has a certain thickness, say dd. From Snell's law about refraction, calling θiθi the angle between the incoming ray and the normal to the surface, nini the refractive index of the medium outside the slab, nmnm the refractive index inside the slab and θmθmthe angle between the ray in the slab and the normal to the surface:

nmsinθm=nisinθinmsin⁡θm=nisin⁡θi

you can derive the angle θmθm:

sinθm=ninmsinθisin⁡θm=ninmsin⁡θi

Then, from basic trigonometry, you know that:

sin2θm+cos2θm=1 ⇒ cosθm=1−sin2θm−−−−−−−−−√sin2⁡θm+cos2⁡θm=1 ⇒ cos⁡θm=1−sin2⁡θm

and, with dd as defined and ll length of the path of the ray inside the slab, you can write:

d=l⋅cosθm ⇒ l=dcosθmd=l⋅cos⁡θm ⇒ l=dcos⁡θm

Calling xx the lateral displacement you are asking for:

x=l⋅sinθmx=l⋅sin⁡θm

By substituting in this equation, you can get your result as an expression of the angle of incidence, the refractive indexes and the thickness of the slab.