Question

Define dispersion without deviation. Derive an expression for its essential condition and resultant dispersion.

Answer 1

Dispersion without deviation: In this combination, the deviation for the mean colour produced by one prism is annulled by the deviation produced for that colour by other prism. However, after passing through the two prisms, there is some dispersion of light left thus kind of combination is used in direct vision spectroscope.
C is crown glass and F is flint glass prism. They are joined with their bases opposite the angles A and A' of these prisms are so chosen that the resultant deviation is zero and there is only dispersion which is called dispersion without deviation.
Essential conditions for dispersion without deviation-The essential condition for it is that deviation produced by crown glass prism is equal to the deviation produced by flint glass prism.
Let the refractive indices of crown and flint glass prisms for mean ray are ${\mu }_{\gamma }$ and ${\mu }_{\gamma }^{\prime }$, then deviation of mean ray by crown prism.
${\delta }_{\gamma }=({\mu }_{\gamma }-1)A$ ......(1)
Deviation of this ray by flint prism
${\delta }_{\gamma }^{\prime }=({\mu }_{\gamma }^{\prime }-1)A^{\prime }$ ........(2)
But resultant deviation is zero, thus
${\delta }_{\gamma }=-{\delta }_{\gamma }^{\prime }$
Substituting values,
$({\mu }_{\gamma }-1)A=-({\mu }_{\gamma }^{\prime }-1)A^{\prime }$
or $A^{\prime }=-\left[\frac{({\mu }_{\gamma }-1)}{({\mu }_{\gamma }^{\prime }-1)}\right]A$ .........(3)
The negative sign shows that the refracting angles of both the prisms are in opposite directions.
Resultant dispersion: Let the refractive indices of crown glass prism for red and violet colour rays are ${\mu }_R$ and ${\mu }_v$ respectively. Similarly for flint glass prism they are ${\mu }_v^{\prime }$ and ${\mu }_R^{\prime }$.
Thus angular dispersion for crown glass prism
$\theta =({\mu }_v-{\mu }_R)A$ ..........(4)
Similarly angular dispersion for flint glass prism
${\theta }^{\prime }=({\mu }_v^{\prime }-{\mu }_R^{\prime })A^{\prime }$ ........(5)
This resultant angular dispersion
$\theta ={\theta }^{\prime }-\theta$
Substituting values $\theta =({\mu }_v^{\prime }-{\mu }_R^{\prime })A^{\prime }-({\mu }_v-{\mu }_R)A$
Putting the value of A' in eqn. (3),
$\theta =({\mu }_v^{\prime }-{\mu }_R^{\prime })\left[\frac{({\mu }_{\gamma }-1)}{({\mu }_{\gamma }^{\prime }-1)}\right]A-({\mu }_v-{\mu }_R)A$
$\theta =({\mu }_{\gamma }-1)A\left[\frac{{\mu }_v^{\prime }-{\mu }_R^{\prime }}{{\mu }_{\gamma }^{\prime }-1}-\frac{{\mu }_{\gamma }-{\mu }_R}{{\mu }_{\gamma }-1}\right]$
$\theta =({\mu }_{\gamma }-1)A({\omega }^{\prime }-\omega )$
$[Here\omega =\frac{{\mu }_v-{\mu }_R}{{\mu }_{\gamma }-1}and{\omega }^{\prime }=\frac{{\mu }_{\gamma }^{\prime }-{\mu }_R^{\prime }}{{\mu }_{\gamma }^{\prime }-1}]$.