Derive the expression for Angle of prism and angle of deviation in terms of angle of incident, angle of emergence and angle.
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Derive the expression for Angle of prism and angle of deviation in terms of angle of incident, angle of emergence and angle.
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Derivation :-
Refer to the attachment for diagram .
Let's suppose that ;
A = Angle of prism
δ = Angle of deviation
μ1 = refractive index of air (μ1 = 1)
μ2 = μ = refractive index of prism
Now,
At the refracting surface AB :
=> μ1•sin(i1) = μ2•sin(i2)
=> sin(i1) / sin(r1) = μ2 / μ1
=> sin(i1) / sin(r1) = μ / 1
=> sin(i1) / sin(r1) = μ -------(1)
Also,
At the refracting surface AC :
=> μ2•sin(r2) = μ1•sin(r1)
=> sin(i2) / sin(r2) = μ2 / μ1
=> sin(i2) / sin(r2) = μ / 1
=> sin(i2) / sin(r2) = μ --------(2)
Now,
In ∆MAN ,
=> ∠A + ∠AMN + ∠ANM = 180°
=> ∠A + (90° - r1) + (90° - r2) = 180°
=> ∠A - r1 - r2 + 180° = 180°
=> ∠A = r1 + r2 -------(3)
Hence,
The angle of prism is given by ;
∠A = r1 + r2
Now,
In ∆PMN ,
=> δ = ∠PNM + ∠PMN
=> δ = (i1 - r1) + (i2 - r2)
=> δ = i1 + i2 - (r1 + r2)
=> δ = i1 + i2 - ∠A ----------(4)
Hence,
The angle of deviation is given by ;
δ = i1 + i2 - ∠A