Angular Dispersion
Angular dispersion produced by a prism for white light is the difference in the angles of deviation of two extreme colors i.e. violet and red colors. In others, it is the angle in which all colors of white light are contained, fig. 1.
If δv = deviation of violet color
δr = deviation of red color
Then, angular dispersion = δv – δr
Now, the deviation δ through a thin prism of refracting angle A is
δ = (µ - 1) A
Therefore, δv = (µv – 1) A (1)
And δr = (µr – 1) A (2)
Subtracting, (2) from (1), we get
δv – δr = (µv – 1)A - (µr – 1)A
= (µv – 1 - µr + 1) A
Therefore, angular dispersion,
(δv – δr) = (µv - µr) A (3)
Obviously, angular dispersion produced by a prism depends upon (i) angle of prism, (ii) nature of material of the prism.
Dispersion power
The dispersion power of a prism is defined as the ratio of angular dispersion to the mean deviation produced by the prism. It is represented by ω.
The mean deviation produced by the prism is
δ = (µ - 1) A
Angular dispersion produced by the prism is
δv – δr = (µv - µr)A
As dispersive power = angular dispersion/mean dispersion
Therefore, ω = δv – δr /δ (4)
ω = µv - µr/µ - 1 = dµ/ µ - 1 (5)
Where µv - µr = dµ (i.e. differences in refractive indices of prism for violet and red colors).
Clearly, ω depends only on nature of material of the prism.
From (4), δv – δr = ω × δ
i.e. angular dispersion = dispersive power × mean deviation
Note: we can suitably combine two prisms of different materials so as to produce;
(i) Dispersion without deviation, and
(ii) Deviation without dispersion.
Dispersion power of a prism depends only on nature of material of the prism. However, angular dispersion and mean deviation, both depends also on angle of prism.
Example: calculate the dispersive power for crown glass from the given data µv = 1.5230, µr = 1.5145.
Solution: µv = 1.5230,
µr = 1.5145 ω =?
Mean refractive index,
µ = µv - µr = (1.5230 + 1.5245)/2
µ = 1.5187
µ = (µv - µr)/(µ - 1)
= (1.5230 – 1.5145)/(1.5187 – 1)
= 0.0085/0.5187
Example: the refractive indices of material of a prism for blue and red colors are 1.532 and 1.514 respectively. Calculate angular dispersion produced by the prism if angle of prism is 8°.
Solution: here, µb = 1.532
And µr = 1.514 A = 8°
Angular dispersion = (µb - µr) A
= (1.532 – 1.514) × 8
= 0.018 × 8
= 0.144°
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