We
have to prove that a ray of light incident on the surface of a sheet
of plate glass of thickness t emerges from the opposite face parallel
to its initial direction but displaced sideways as shown in the
figure. (a) We have to
show that, for small
angles of incidence
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Solution: Click For PDF Version (a)
Let n
be the refractive index of the plate glass. We assume that the angle
of incidence
The angle of
refraction
Using the
where
The thickness of the plate glass sheet is t. Therefore, the separation between the points where the refracted ray meets the opposite edge of the sheet and the point where the unrefracted incident ray would have touched the opposite edge of the sheet will be given by
In the limit of small angles, the displacement x will also be approximately equal to d. Hence
As the front and the back faces of the sheet are parallel, from the line diagram of the figure and the use of Snell’s law for refraction from glass to air and from air to glass, we note that the ray will emerge from the glass sheet parallel to the incident ray. (b)
Refractive
index of crown glass,
The thickness
of the crown glass sheet,
Therefore, the displacement of the incident ray due to refraction in the glass sheet will be
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,
this displacement is given by
angle of incidence through a
1.0-cm thick sheet of crown glass.
.
is
given by the Snell’s law
,
.
-coordinate
system shown in the figure, we can write for the equations of
straight lines for the incident ray and the refracted rays the
following equations:
is the distance inside the plate glass measured from the edge where
the incident ray meets the glass sheet. In the small angle
approximation these equations can be written as
.