Deviation and Dispersion Formulas

A → Angle of prism
δ → Angle of deviation
δ = i + e – A
A = r₁ + r₂

1. Condition of no emergence

µ > \(\frac{1}{\sin A / 2}\) > cosec(A/2)
Condition of Grazing Emergence (means = e = 90°)
r₂ = θ_c
i = sin^-1 [\(\sqrt{\mu^{2}-1}\) sin A – cos A]
Condition of max. deviation
Deviation will be max. when i_max = 90°
δ_max = 90° + sin^-1 [µ sin (A – θ_c)] – A
Condition for min. deviation
it occur when i = e, r₁ = r₂ = r & A = r/2
δ_min = (2i – A)

2. Prism formula

µ = \(\frac{\sin \left(\frac{\delta_{\min }+A}{2}\right)}{\sin A / 2}=\frac{\sin \left(\frac{\delta_{\min }+A}{2}\right)}{\sin A / 2}\)
For thin prism
deviation δ = A(µ – 1)

3. Dispersion

Splitting of a beam of white light into its constituent colours.

4. Cauchy’s formula

µ = A + \(\frac{C}{\lambda^{4}}+\frac{C}{\lambda^{4}}\)
A, B & C are constants.

5. Angular dispersion

θ = δ_v – δ_r = (µ_v – µ_r) A

6. Dispersive power

\(\frac{\theta}{\delta_{y}}=\omega=\frac{\mu_{\mathrm{v}}-\mu_{\mathrm{r}}}{\mu_{\mathrm{y}}-1}=\frac{\Delta \mu}{\mu-1}\)

7. Conbination of prisms

net deviation
δ = δ₁ – δ₂
δ = (µ₁ – 1) A₁ – (µ₂ – 1) A₂
net angular dispersion
δ_v – δ_r = (µ_1v – µ_1r) A₁ – (µ_2v – µ_2r) A₂
= (µ_1y – 1)ω₁A₁ – (µ_2y – 1)ω₂A₂

8. Dispersion without deviation

δ_y = o, (µ_1y – 1) A₁ = (µ_2y – 1)A₂
δ_v – δ_r = δ₁ (ω₁ – ω₂) = δ₂ (ω₁ – ω₂)

9. Average deviation without dispersion

δ_v – δ_r = 0
(µ_1y – 1) ω₁A₁ = (µ_2y – 1) ω₂A₂
δ = δ₁\(\left(1-\frac{\omega_{1}}{\omega_{2}}\right)\)
δ₁ = (µ_1y – 1) A₁

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