Question

Using the Huygens' principle, briefly describe reflection of a plane wavefront from a reflecting surface. Hence, prove the laws of reflection.

Hint:

Draw wavefront AB, wavelet from A of radius BC, and show triangles ABC and AEC are congruent.

Answer 1

Step 1: Huygens' principle:
Every point of a wavefront acts as a source of secondary spherical wavelets that spread with the speed of the wave. The new wavefront is the common tangent (envelope) to these wavelets after a time \(\tau\).

Step 2: Set-up for reflection (diagram described):
Let MN be a plane reflecting surface and AB a plane wavefront of a parallel beam, incident at an angle \(i\) to the surface. The incident rays are perpendicular to AB, and the angle between the incident ray and the normal is the angle of incidence \(i\).
The wavefront touches the surface first at A. Point B is at a distance BC from the surface, where C is the point on MN that B reaches. The speed of the wave is \(v\) and it stays in the same medium.

Step 3: Draw the reflected wavefront:
The disturbance from B reaches C in time \(\tau = \dfrac{BC}{v}\). In the same time the wavelet from A, on reflection, spreads out as a hemisphere of radius \(AE = v\tau = BC\).
Draw the tangent from C to this hemisphere. It touches it at E. Then CE is the reflected wavefront, and the line AE is perpendicular to CE: it is the reflected ray.

Step 4: Prove the laws:
Compare the triangles ABC and AEC. Both are right-angled (\(\angle ABC = \angle AEC = 90^{\circ}\)). They share the side AC, and \(BC = AE = v\tau\). So the triangles are congruent, and \(\angle BAC = \angle ECA\).
The angle \(\angle BAC\) equals the angle of incidence \(i\), because the sides of the angle are perpendicular to the incident ray and to the normal. In the same way, \(\angle ECA\) equals the angle of reflection \(r\). Hence
\[ i = r \]
This is the second law: the angle of incidence equals the angle of reflection.
The incident ray, the reflected ray and the normal at the point of incidence all lie in the plane of the figure. This is the first law of reflection.

Final Answer:
Both laws of reflection follow from Huygens' construction: the incident ray, reflected ray and normal are in one plane, and the angle of incidence equals the angle of reflection.\[ \boxed{i = r} \]