MP Board · Class 12 · Physics · Wave OpticsHuygens का तरंग सिद्धांत लिखिए। इस सिद्धांत के आधार पर तरंग सिद्धांत के अनुसार प्रकाश के परावर्तन के नियमों (आपतन कोण परावर्तन कोण के बराबर होता है) को सिद्ध कीजिए।

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. According to this principle:

1. Every point on a given wavefront acts as a source of secondary wavelets, sending out disturbances in all directions with the velocity of light in that medium.
2. The new position of the wavefront at any subsequent time is given by the envelope of these secondary wavelets touching them tangentially in the forward direction.

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

• Let $v$ be the speed of light in the medium.
• Let $t$ be the time taken by the disturbance from point $B$ to reach point $C$ on the reflecting surface. Therefore, distance $BC = vt$.
• According to Huygens' principle, as the wavefront touches point $A$ first, secondary wavelets start originating from $A$ and spread out with radius $vt$ in time $t$.
• With $A$ as center and radius $AD = vt$, an arc is drawn. From point $C$, a tangent $CD$ is drawn to this arc. Thus, $CD$ represents the reflected wavefront.

Geometric Derivation\nIn $\triangle ABC$ and $\triangle ADC$:

• $\angle ABC = \angle ADC = 90^\circ$ (since wavefront is perpendicular to the rays)
• Side $AC$ is common to both triangles.
• $BC = AD = vt$ (since distance covered in time $t$ with speed $v$ is equal) \nTherefore, by RHS (Right angle-Hypotenuse-Side) congruence criterion, $\triangle ABC \cong \triangle ADC$. \nFrom the congruence of triangles, the corresponding angles are equal: $$\angle BAC = \angle DCA$$ \nLet the angle of incidence be $i$ and angle of reflection be $r$. From geometry, it can be easily shown that:
• Angle of incidence, $i = 90^\circ - \angle BAC$
• Angle of reflection, $r = 90^\circ - \angle DCA$ \nSince $\angle BAC = \angle DCA$, we get: $$i = r$$ \nThis proves the first law of reflection. Furthermore, the incident wavefront, the normal, and the reflected wavefront all lie in the same plane, which proves the second law of reflection.