Question

Derive the laws of reflection of light using Huygens’ principle.

Explain reflection of light at a plane reflecting surface on the basis of Huygen's principle.

Answer 1

Reflection of a plane wavefront of light at a plane surface

Where MN: Plane mirror,
RA and QC: Incident rays, 
AP: Normal to MN,
AB: Incident wavefront,
i: Angle of the incident,
CE: Reflected wavefront,
r: Angle of reflection

When wavefront AB is incident on the mirror, at first, point A becomes a secondary source and emits secondary waves in the same medium. If T is the time taken by the incident wavefront to travel from B to C, then BC = vT. During this time, the secondary wave originating at A covers the same distance, so that the secondary spherical wavelet has a radius vT at time T.

To construct the reflected wavefront, a hemisphere of radius vT is drawn from point A. Draw a tangent EC to the secondary wavelet.

The arrow AE shows the direction of propagation of the reflected wave.

AP is normal to MN at A.

∠RAP = i = angle of incidence and

∠PAE = r = angle of reflection

In ΔABC and ΔAEC,

AE = BC and ∠ABC = ∠AEC = 90°

∴ ΔABC and ΔAEC are congruent.

∴ ∠ACE = ∠BAC = i        .....(1)

 Also, as AE is perpendicular to CE and AP is perpendicular to AC,

∠ACE = ∠PAE = r       .....(2)

∴ From Eqs (1) and (2),

i = r

Thus, the angle of incidence is equal to the angle of reflection. This is the first law of reflection. Also, it can be seen from the figure that the incident ray and reflected ray lie on the opposite sides of the normal to the reflecting surface at the point of incidence and all of them lie in the same plane. This is the second law of reflection. Thus, the laws of reflection of light can be deduced by Huygens' construction of a plane wavefront.