Question

The index of refraction for violet light in silica flint glass is 1.66 and that for red light is 1.62. What is the angular dispersion of visible light passing through an equilateral prism of apex angle 60.0 degree if the angle of incidence is 47.0 degree? (See figure. The index of refraction for air is 1.00.) Your response is within 10% of the correct value. This may be due to roundoff error, or you could have a mistake in your calculation. Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error.

Answer 1

According to Snell’s law -

n₁ sin θ₁ = n₂ sin θ₂ given, θ₁ = 47.0°
For the refraction of violet light ( n = 1.66 ) from air to glass

So, ( 1.00 ) sin 47.0° = ( 1.66 ) sin θ₂₁

=> sin θ₂₁ = ( 1.00 / 1.66 ) sin 47.0°
=> θ₂₁ = sin ⁻ ¹ [ ( 1.00 / 1.66 ) sin 47.0° ] = 26.1°

Now, for the refraction of violet light from glass to air
n₂ sin θ₂ = n₃ sin θ₃₁

heree, n₂ = 1.66, θ₂ = 60 - 26.1° =33.9^o , n₃ = 1.00
=> sin θ₃₁ = ( n₂ / n₃ ) sin θ₂ = n₂ sin θ₂₂ = ( 1.66 ) sin 33.9°
… θ₃₁ = sin ⁻ ¹ [ ( 1.66 ) sin 33.9° ] = 67.8°

Again, for the refraction of red light ( n = 1.62 ) from air to glass -

=> ( 1.000 ) sin 47° = ( 1.62 ) sin θ₂

=> sin θ₂ = ( 1.000 / 1.62 ) sin 47°
=> θ₂ = sin ⁻ ¹ [ ( 1.000 / 1.62 ) sin 47° ] = 26.8°

For the refraction of red light from glass to air -

n₂ sin θ₂ = n₃ sin θ₃₂ given that, n₂ = 1.62, θ₂ = 26.8°, n₃ = 1.00
=> sin θ₃₂ = ( n₂ / n₃ ) sin θ₂ = n₂ sin θ₂ = ( 1.62 ) sin 26.8°
=> θ₃₂ = sin ⁻ ¹ [ ( 1.62 ) sin 26.8° ] = 47°

Therefore, γ = θ₃₁ - θ₃₂ = 67.8 - 47 = 20.8°