Question

State Huygens’ Principle. Use it to prove the laws of reflection or laws of refraction at a plane surface.

Hint:

Huygens’ principle explains wave behavior geometrically. Reflection: \(i = r\) Refraction: \(n_1 \sin i = n_2 \sin r\) It proves light behaves as a wave.

Answer 1

Concept:
Huygens’ principle explains wave propagation using secondary wavelets. The new wavefront at any time is the envelope (tangent surface) of these secondary wavelets. \[ \text{New wavefront} = \text{Envelope of secondary wavelets} \] This principle helps derive reflection and refraction laws geometrically.
Step 1: Statement of Huygens’ Principle
Every point on a wavefront behaves like a source of secondary spherical wavelets. The forward envelope of these wavelets gives the new wavefront.
Step 2: Reflection using Huygens’ Principle
Consider a plane wavefront incident on a plane mirror at an angle \(i\).
- Let AB be the incident wavefront.
- Point A touches the mirror first.
- After time \(t\), point B reaches the mirror.
During this time, a secondary wavelet from A spreads as a circle (in 2D).
Step 3: Construct reflected wavefront
Draw a tangent from point B to the secondary wavelet from A.
This tangent gives the reflected wavefront.
From geometry of construction: \[ \angle i = \angle r \] Thus, angle of incidence equals angle of reflection. (OR) Refraction using Huygens’ Principle
Step 4: Refraction setup
Let a wavefront travel from medium 1 to medium 2 with speeds \(v_1\) and \(v_2\).
- Point A enters second medium first
- Point B still in first medium
Wavelets from A travel slower or faster depending on medium.
Step 5: Construct refracted wavefront
After time \(t\):
- Distance traveled by A = \(v_2 t\)
- Distance traveled by B = \(v_1 t\)
Using right triangle geometry: \[ \frac{\sin i}{\sin r} = \frac{v_1}{v_2} \] Using refractive index relation: \[ n = \frac{c}{v} \] We get: \[ n_1 \sin i = n_2 \sin r \] This is Snell’s law of refraction.