Derive the formula connecting the refractive index of a prism and the angle of mean deviation. If the angle of the prism is A.
The Correct Answer and Explanation is:
To derive the formula connecting the refractive index nnn of a prism and the angle of mean deviation Δ\DeltaΔ, we begin with the general principles of light refraction in a prism.
Key Variables:
- AAA: Angle of the prism (angle between the two faces of the prism).
- nnn: Refractive index of the prism.
- Δ\DeltaΔ: Mean deviation (the average angular deviation of the light from its original path after passing through the prism).
- θ1\theta_1θ1: Angle of incidence at the first face of the prism.
- θ2\theta_2θ2: Angle of refraction inside the prism.
- θ3\theta_3θ3: Angle of incidence at the second face of the prism.
Refraction through the Prism:
When light passes through a prism, it undergoes two refractions:
- At the first surface: The angle of incidence is θ1\theta_1θ1, and the light refracts into the prism at an angle θ2\theta_2θ2. Using Snell’s law: n1sin(θ1)=nsin(θ2)n_1 \sin(\theta_1) = n \sin(\theta_2)n1sin(θ1)=nsin(θ2) where n1=1n_1 = 1n1=1 (since the refractive index of air is 1).
- At the second surface: The light exits the prism, and the angle of incidence is θ3\theta_3θ3, while the angle of refraction in air is θ4\theta_4θ4. Applying Snell’s law again: nsin(θ3)=n1sin(θ4)n \sin(\theta_3) = n_1 \sin(\theta_4)nsin(θ3)=n1sin(θ4) where n1=1n_1 = 1n1=1 again, so: sin(θ3)=sin(θ4)\sin(\theta_3) = \sin(\theta_4)sin(θ3)=sin(θ4)
Angle of Deviation:
The total angle of deviation Δ\DeltaΔ is the angular change in the direction of the light as it passes through the prism. It can be expressed as:Δ=(θ1+θ3)−A\Delta = (\theta_1 + \theta_3) – AΔ=(θ1+θ3)−A
where AAA is the angle of the prism.
Relation Between Deviation and Refractive Index:
To derive the relationship, we use the formula for the deviation for small angles:Δ=(n−1)A\Delta = (n – 1) AΔ=(n−1)A
Thus, the mean deviation Δ\DeltaΔ is directly proportional to the refractive index nnn and the angle of the prism AAA.
Conclusion:
The formula connecting the refractive index nnn of a prism and the angle of mean deviation Δ\DeltaΔ is:Δ=(n−1)A\Delta = (n – 1) AΔ=(n−1)A
This formula indicates that the mean deviation increases as the refractive index or the angle of the prism increases.
