Refraction of Light Through Prism
Ordinary white light, as we know, is a superimposition of waves having wavelength throughout the visible spectrum
Prism is that optical instrument which splits this white light into its constituent wavelengths
A prism dispersing white light into its composite wavelengths or colours
This breaking up of white light inside a prism happens through refraction at the prism-air interface
Let us now understand some mathematical relations incorporating Refraction of light through a prism
We consider the principal section of a prism,
$ABC$
Let two normals
$N_{1}M$
and
$N_{2}M$
be incident on the interfaces
$AB$
and
$AC$
respectively
Let a ray
$PQ$
be incident on
$AB$
making an incident angle
$i$
As the ray is travelling from rarer to denser medium; it bends towards the normal
Then it refracts at
$AC$
and bends away from the normal (denser to rarer) forming the emergent ray
This emergent ray forms an angle
$e$
with the normal at
$AC$
known as the angle of emergence
The angles of refraction are
$r_{1}$
and
$r_{2}$
respectively
Now if the emergent ray is extended backwards to
$O$
, then we get the angle of deviation, denoted by
$δ$
$δ$
tells, to what extent the ray PQ had bent from its original path PQOR to OS, owing to refraction by the prism
Now consider
$△OQT$
Let,
$∠OQT=α,∠OTQ=β$
$∴δ=α+β(Sum of Int angles=Ext angle)$
Now from the diagram, we can write the interior angles as
$α+r_{1}=i$
$∴α=i−r_{1}$
Similarly
$β=e−r_{2}$
Now simplifying the equation
$δ=α+β$
we get
$δ=i−r_{1}+e−r_{2}$
$→δ=i+e−(r_{1}+r_{2})$
Now let us consider quadrilateral
$AQMT$
$∠AQM+∠ATM=180_{0}(MQ and MT are normals)$
So, we can say quadrilateral
$AQMT$
is cyclic
$∴∠A+∠M=180_{0}→Eqn 1$
In
$△MQT$
$∠r_{1}+∠r_{2}+∠M=180→Eqn 2$
From Eqn 1 and 2 we get,
$∠r_{1}+∠r_{2}+∠M=∠M+∠A$
$∴∠r_{1}+∠r_{2}=∠A$
As
$A=r_{1}+r_{2}$
and
$δ=i+e−(r_{1}+r_{2})$
$∴δ=i+e−A$
$→δ+A=i+e$
Thus we get a relation between the angle of deviation
$δ$
, the angle of prism
$A$
, the angle of incidence
$i$
and the angle of emergence
$e$
Revision
The relation between
$δ$
and
$A$
is
$δ+A=i+e$
also
$A$
can be written as
$A=r_{1}+r_{2}$
The relation
$δ+A=i+e$
relates between the angle of deviation
$δ$
, the angle of prism
$A$
, the angle of incidence
$i$
and the angle of emergence
$e$
The End