Define angular dispersion for prism. Obtain its expression for thin prism. Relate it with the refractive indices of the material of the prism for corresponding colours.
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In a prism, the angle of deviation (δ) decreases with increase in the angle of incidence (i) up to a particular angle. This angle of incidence where the angle of deviation in a prism is minimum is called the minimum deviation position of the prism and that very deviation angle is known as the minimum angle of deviation (denoted by δmin, Dλ, or Dm).
Light is deflected as it enters a material with refractive index > 1.
A ray of light is deflected twice in a prism. The sum of these deflections is the deviation angle.
When the entrance and exit angles are equal, the deviation angle of a ray passing through a prism will be minimal.
The angle of minimum deviation is related with the Refractive index as:
{\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}{\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}
This is useful to calculate the refractive index of a material. Rainbow and halo occur at minimum deviation. Also, a thin prism is always set at minimum deviation.
In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism.[1][2][3] Also, the angles of refractions are equal i.e. r1 = r2. And, the angle of incidence and angle of emergence equal each other (i = e). This is clearly visible in the graph below.
The formula for minimum deviation can be derived by exploiting the geometry in the prism. The approach involves replacing the variables in the Snell's law in terms of the Deviation and Prism Angles by making the use of the above properties.
Minimum Deviation.jpg
From the angle sum of {\textstyle \triangle OPQ}{\textstyle \triangle OPQ},
{\displaystyle A+\angle OPQ+\angle OQP=180^{\circ }}{\displaystyle A+\angle OPQ+\angle OQP=180^{\circ }} {\displaystyle \implies A=180^{\circ }-(90-r)-(90-r)}{\displaystyle \implies A=180^{\circ }-(90-r)-(90-r)} {\displaystyle \implies r={\frac {A}{2}}}{\displaystyle \implies r={\frac {A}{2}}}
Using the exterior angle theorem in {\textstyle \triangle PQR}{\textstyle \triangle PQR},
{\displaystyle D_{m}=\angle RPQ+\angle RQP}{\displaystyle D_{m}=\angle RPQ+\angle RQP} {\displaystyle \implies D_{m}=i-r+i-r}{\displaystyle \implies D_{m}=i-r+i-r} {\displaystyle \implies 2r+D_{m}=2i}{\displaystyle \implies 2r+D_{m}=2i} {\displaystyle \implies A+D_{m}=2i}{\displaystyle \implies A+D_{m}=2i} {\displaystyle \implies i={\frac {A+D_{m}}{2}}}{\displaystyle \implies i={\frac {A+D_{m}}{2}}}
This can also be derived by putting i = e in the prism formula: i + e = A + δ
From Snell's law,
{\displaystyle n_{21}={\dfrac {\sin i}{\sin r}}}{\displaystyle n_{21}={\dfrac {\sin i}{\sin r}}}
{\displaystyle \therefore n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}
{\displaystyle \therefore n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}
[4][3][1][2][5][excessive citations]
{\displaystyle \therefore D_{m}=2\sin ^{-1}\left(n\sin \left({\frac {A}{2}}\right)\right)-A}
{\displaystyle \therefore D_{m}=2\sin ^{-1}\left(n\sin \left({\frac {A}{2}}\right)\right)-A}
(where n is the refractive index, A is the Angle of Prism and Dm is the Minimum Angle of Deviation.)
This is a convenient way used to measure the refractive index of a material(liquid or gas) by directing a light ray through a prism of negligible thickness at minimum deviation filled with the material or in a glass prism dipped in it
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