MP Board · Class 12 · Physics · Wave OpticsState Huygens' Principle. Using Huygens' construction, verify the laws of reflection of light (angle of incidence equals angle of reflection) with a neat and labeled ray diagram.

Statement of Huygens' Principle\nHuygens' Principle is a geometrical construction used to determine the position of a wavefront at any later time, given its position at any instant. It states that:

1. Every point on a given wavefront acts as a source of secondary spherical wavelets, which spread out in all directions with the speed of light in that medium.
2. The forward envelope of these secondary wavelets at any later time gives the new position of the wavefront.

Verification of Laws of Reflection\nTo verify the laws of reflection using Huygens' principle, let us consider a plane wavefront $AB$ incident obliquely on a reflecting surface (plane mirror) $XY$ at an angle of incidence $i$.

• Construction of Reflected Wavefront:

  • Let $v$ be the speed of light in the medium and $t$ be the time taken by the wave front to travel from point $B$ to point $C$.
  • Therefore, the distance $BC = vt$.
  • As point $A$ hits the surface first, it begins to emit secondary wavelets. By the time point $B$ reaches $C$, the secondary wavelets from $A$ will have spread out to a radius $vt$.
  • With $A$ as the center and radius $AD = vt$, draw an arc representing the secondary wavelet.
  • Draw a tangent $CD$ from point $C$ to this arc. Here, $CD$ represents the reflected wavefront.

• Proof of Angle of Reflection:

  • Consider $\triangle ABC$ and $\triangle ADC$:
    • $BC = AD = vt$ (Distance traveled by light in time $t$)
    • $\angle ABC = \angle ADC = 90^\circ$ (Wavefront is perpendicular to the rays)
    • $AC = AC$ (Common hypotenuse)
  • Therefore, by RHS congruence criterion, $\triangle ABC \cong \triangle ADC$.
  • Consequently, $\angle BAC = \angle DCA$.

• Conclusion:

  • Let the normal to the surface at the point of incidence be drawn. The angle of incidence $i$ is the angle between the incident wavefront and the reflecting surface, which equals $\angle BAC$. Similarly, the angle of reflection $r$ equals $\angle DCA$.
  • Since $\angle BAC = \angle DCA$, we get $i = r$, which is the second law of reflection.
  • Furthermore, the incident ray, reflected ray, and normal all lie in the same plane, satisfying the first law of reflection.