Making use of the ideas of equal transit times between corresponding points and the orthogonality of rays and wavefronts, derive the Law of Reflection and Snell’s Law. The ray diagram of Fig. P.4.32
The Correct Answer and Explanation is :
Derivation of the Law of Reflection and Snell’s Law
Using Fermat’s principle, which states that light travels the path that requires the least time, we derive the Law of Reflection and Snell’s Law using wavefronts and rays.
1. Law of Reflection
Consider a plane wavefront approaching a reflecting surface at an angle. The wavefront meets the surface at points A and B. If the wavefront is to maintain equal transit times between corresponding points, the new wavefront after reflection must form such that the angles are equal.
From the diagram:
- The incident wavefront meets the surface at A first and then at B.
- The reflected wavefront must maintain equal distances traveled for light from A and B to form a coherent front.
By considering the orthogonality of rays (perpendicular to wavefronts), we see:
- The angle of incidence θi is equal to the angle of reflection θr, as the path lengths remain equal.
Thus, θi = θr, proving the Law of Reflection.
2. Snell’s Law
When light travels between two media with refractive indices n₁ and n₂, the wavefront changes speed. Let the wavefront strike the interface at A and B with speed v₁ in medium 1 and v₂ in medium 2.
Using equal transit time principles:
- Distance traveled in medium 1: v₁t.
- Distance traveled in medium 2: v₂t.
- The wavefront must remain continuous, so the path difference forms a new refracted wavefront.
From the geometry:
[
\frac{\sin θ_1}{v_1} = \frac{\sin θ_2}{v_2}
]
Since ( v = \frac{c}{n} ), we obtain Snell’s Law:
[
n_1 \sin θ_1 = n_2 \sin θ_2
]
which describes the bending of light.
Now, I will generate the image illustrating this concept.
Here is the generated diagram illustrating the derivation of the Law of Reflection and Snell’s Law using wavefronts and rays. Let me know if you need any modifications or further clarifications!
