State Huygens’ Principle. Use it to prove the laws of reflection or laws of refraction at a plane surface.
Concept Overview:
Huygens’ principle explains how wavefronts propagate. According to this principle, every point on a wavefront acts as a source of secondary wavelets. The new position of the wavefront at a later time is obtained by drawing the forward envelope (common tangent surface) of these secondary wavelets. \[ \text{New wavefront} = \text{Envelope of secondary wavelets} \] This geometrical construction helps in deriving the laws of reflection and refraction.
Step 1: Statement of Huygens’ Principle
Each point on a given wavefront behaves like a source of secondary spherical wavelets. The new wavefront after time \( t \) is the surface tangent to these wavelets in the forward direction.
Step 2: Reflection Using Huygens’ Principle
Consider a plane wavefront incident on a plane mirror at an angle \( i \). - Let AB represent the incident wavefront.
- Point A touches the mirror first.
- After time \( t \), point B reaches the mirror.
During this interval, a secondary wavelet originates from point A and spreads outward.
Step 3: Construction of the Reflected Wavefront
From point B, draw a tangent to the secondary wavelet originating from A. This tangent represents the reflected wavefront. From the geometrical construction: \[ \angle i = \angle r \] Hence, the angle of incidence equals the angle of reflection.
Refraction Using Huygens’ Principle
Step 4: Refraction Setup
Let a wavefront travel from medium 1 to medium 2 with velocities \( v_1 \) and \( v_2 \). - Point A enters the second medium first.
- Point B remains in the first medium.
Since wave speed changes in the second medium, the secondary wavelets travel different distances.
Step 5: Construction of the Refracted Wavefront
After time \( t \): - Distance traveled by A in medium 2 = \( v_2 t \)
- Distance traveled by B in medium 1 = \( v_1 t \)
Using the geometry of the constructed right triangle: \[ \frac{\sin i}{\sin r} = \frac{v_1}{v_2} \] Using the relation between refractive index and speed: \[ n = \frac{c}{v} \] We obtain: \[ n_1 \sin i = n_2 \sin r \] This is Snell’s law of refraction.