Q. A ray of light travelling in air is incident at grazing
angle (Incident angle = 90) on a long rectangular slab of
a transparent medium of thickness t = 1.0 m, The point of
incidence is the origin A(0,0). The medium has a variable
index of refraction n (y) given by
where k = 1.0
The refractive index of air is 1.0.
(a) Obtain a relation between the slope of the trajectory of
the ray at a point B (x, y) in the medium and the incident
angle at that point.
(b) Obtain an equation for the trajectory y(x) of the ray in the medium.
(c) Determine the coordinates () of the point P, where
the ray intersects the upper surface of the slab-air
boundary.
(d) Indicate the path of the ray subsequently.

 5085  207 IIT JEEIIT JEE 1995 Report Error

Solution:

(a) Refractive index is a function of . It varies along y-axis i.e. the boundary separating two media is parallel to -axis or normal at any point will be parallel to -axis.
Secondly, refractive index increases as is increased. Therefore, ray of light is travelling from rarer to denser medium i.e. it will bend towards the normal and shape of its trajectory will be as shown below.
Now, refer to figure (a)
Let be the angle of incidence at any point on its path
or
or slope
(b) but tan
... (i)
Applying Snell's law at and
because
because (Grazing incidence)

because

or

or ... (ii)
Equating Eqs. (i) and (ii), we get
or
or
or ... (iii)
The required equation of trajectory is
(c) At point , where the ray emerges from the slab

[From Eq. (iii)]
Therefore, coordinates of point P are

(d) As and as
Therefore, i.e. the ray will emerge parallel to the boundary at i.e. at grazing emergence.

Solution Image Solution Image