CBSE · Class 12 · Physics · Wave OpticsState Huygens' Principle. Using Huygens' construction, verify the laws of reflection of light (angle of incidence equals angle of reflection).

Introduction to 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 earlier time. It is based on the following fundamental postulates:

• Every point on a given wavefront acts as a source of secondary wavelets, sending out disturbances in all directions with the wave velocity.
• The new wavefront at any subsequent time is the envelope tangent to all these secondary wavelets moving in the forward direction.

Derivation of Laws of Reflection\nLet us consider a plane wavefront $AB$ incident obliquely on a plane reflecting surface $XY$ at an angle of incidence $i$.

1. Construction of Wavefront:

   • Let the wavefront strike the surface at point $A$ first and then at point $B$ after a time interval $t$.
   • Let $v$ be the speed of light in the medium. The time taken by the disturbance to travel from $B$ to $B'$ is given by $t = \frac{BB'}{v}$, so $BB' = vt$.
   • According to Huygens' principle, from point $A$, secondary wavelets start spreading out in the same medium with speed $v$. In time $t$, the secondary wavelets from $A$ will cover a distance $vt$ and form a spherical wave arc of radius $AD = vt$.
   • From point $B'$, we draw a tangent $B'D$ to this arc. Here, $B'D$ represents the reflected wavefront.

2. Proof of Angle of Incidence equals Angle of Reflection:

   • Consider triangle $ABB'$ and triangle $ADB'$.
     • $\angle ABB' = \angle ADB' = 90^\circ$ (by construction)
     • Side $AB' = AB'$ (common hypotenuse)
     • Side $BB' = AD = vt$ (radii of wavelets / distance travelled in time $t$)
   • Therefore, by the RHS (Right angle-Hypotenuse-Side) congruence criterion, $\triangle ABB' \cong \triangle ADB'$.
   • Consequently, the corresponding angles are equal: $\angle BAB' = \angle AB'D$.
   • Since the incident ray and reflected ray are normal to the incident and reflected wavefronts respectively, it can be easily established that the angle of incidence $i$ and angle of reflection $r$ are related as: $$i = r$$

3. Conclusion:

   • Also, the incident wavefront, normal to the reflecting surface, and the reflected wavefront all lie in the same plane, which verifies the second law of reflection. Thus, the laws of reflection are successfully verified using Huygens' wave theory of light.