Name: Derivation of Angle of Deviation by a Prism
Uploaded: 2019-07-19
Duration: 10 min 39 s
Description: Let’s learn how to derive a formula for the angle of deviation and what the condition for minimum deviation should be. Watch this video to learn more.

Question

For any prism, prove that:

n = \frac{\sin (\frac{A + δ m}{2})}{\sin (\frac{A}{2})}

where the terms have their usual meaning.

Accepted Answer

$∠ O F G = i_{1} - r_{1}$ and $∠ O G F = i_{2} - {r_{2}}_{}$

$∆ F O G, δ$ is the exterior angle.

$δ = ∠ O F G + ∠ O G F$

$= (i_{1} - r_{1}) +$ $(i_{2} - r_{2})$

$= (i_{1} + i_{2}) - (r_{1} + r_{2})$ ----- (i)

The normal $F N_{1}$ and $G N_{2}$ on faces $P Q and P R$ respectively when produced, meet at $N$ .

Let $∠ F N G = θ$

So, $r_{1} + r_{2} + θ = 180 °$ -------- (ii)

In quadrilateral $P F N G$ , $∠ P F N = 90 °$

$\Rightarrow A + 90 ° + θ + 90 ° = 360 ° \Rightarrow A + θ = 180 °$ -----------(iii)

Comparing ii and iii, $r_{1} + r_{2} = A$ ------- (iv)

Putting this in eq (i) , we get $δ = i_{1} + i_{2} - A$

For, a very thin prism, we know that

, $i_{1} = i_{2} = i r_{1} = r_{2} = r$

From eq (iv),

$r + r = A \Rightarrow r = \frac{A}{2}$

and, $i + i = A = δ_{m i n}$

$\Rightarrow i = \frac{A + δ_{m i n}}{2}$

Using snell's law, $μ = \frac{\sin (i)}{\sin (r)}$

Substituting values, we get, $μ = \frac{\sin (\frac{A + δ}{2})}{\sin (\frac{A}{2})}$

For a very thin prism, $μ = \frac{\frac{A + δ}{2}}{\frac{A}{2}}$

$\Rightarrow μ A = a + δ$ $\Rightarrow A (μ - 1) = δ$

Deviation does not depend on the angle of incidence for a very thin prism.

For any prism, prove that:
$n = \frac{\sin (\frac{A + δ m}{2})}{\sin (\frac{A}{2})}$
where the terms have their usual meaning.

Important Questions on Refraction and Dispersion of Light through a Prism

Learn and Prepare for any exam you want !

For any prism, prove that: n=sin(A+δm2)sin(A2) where the terms have their usual meaning.

Important Questions on Refraction and Dispersion of Light through a Prism

Learn and Prepare for any exam you want !

For any prism, prove that:
$n = \frac{\sin (\frac{A + δ m}{2})}{\sin (\frac{A}{2})}$
where the terms have their usual meaning.