Question

Derive the law of reflection using Huygen’s Wave Theory. 

Answer 1

Huygen’s wave theory :

Reflection at a plane surface : Consider a plane reflecting surface XY. Let AB be a plane wavefront of light incident obliquely on XY. When the incident wavefront touches XY at A, a secondary wavelet starts, spreading from A according to Huygens’ principle. Let the ray at B reach XY at D after a time t. If v is the speed of light in air then BD = vt. During this time t, the secondary wavelet from A spreads over a hemisphere of radius vt. with centre at A. Let CD be a tangent to this hemisphere. Then AC = BD. C and D are in the same phase. If we consider all the points between A and D, then CD will be tangential to all the secondary wavelets originating from these points at the end of t seconds. Hence CD is the reflected wavefront.

Draw AN normal to XY. Then 

${\displaystyle \angle PAN=i}$ , the angle of incidence , and ${\displaystyle \angle NAC=r}$ the angle of reflection

In triangles BAD and CDA

AC = BD = vt; AD is common, and ${\displaystyle \angle ABD=\angle ACD={90}^{\circ }}$ because the rays are normal to wavefronts.

${\displaystyle \therefore }$ Triangles BAD and CDA are congruent

${\displaystyle \therefore \angle DAC=\angle BDA,{90}^{\circ }-r={90}^{\circ }-i}$

OR  ${\displaystyle \therefore i=r}$

i.e , angle of incidence is equal to angle of reflection .