9 min readPathway sheet3-mark proof with a Huygens' construction diagram; sometimes asked with refraction as a 5-mark question

Reflection of a plane wave: proving the law of reflection

Using Huygens' wavelets on a mirror, the two triangles formed are congruent, so the angle of incidence equals the angle of reflection.

Law of reflectioni = r; the incident ray, the reflected ray and the normal lie in one plane

Key equalityBC = AE = vτ, because incident and reflected light travel in the same medium

Congruent trianglesTriangles ABC and AEC: AC common, BC = AE, right angles at B and E (RHS)

On reflectionSpeed, frequency and wavelength all stay the same

Proof of the law of reflection

A plane wavefront AB falls on a plane reflecting surface MN at an angle i. Light travels at speed v in the medium.

1
End A of the wavefront touches the surface first. End B still has to travel a distance BC to reach the surface at C.
2
Let this take a time τ. Then BC = vτ.
3
In the same time τ, a secondary wavelet from A spreads back into the same medium. Its radius is AE = vτ.
4
From C, draw a tangent CE to this wavelet, touching it at E. CE is the reflected wavefront.
5
In triangles ABC and AEC: AC is common, BC = AE = vτ, and ∠ABC = ∠AEC = 90°.
6
So the two triangles are congruent (RHS rule). Hence ∠BAC = ∠ECA.
7
∠BAC is the angle of incidence i, and ∠ECA is the angle of reflection r.
The angle between a wavefront and the surface equals the angle between the ray and the normal, because both pairs are at 90°.
8★
So, i = r: the angle of incidence equals the angle of reflection.

★ key step

How to draw the diagram for reflection

Draw a horizontal line MN; label it reflecting surface.
Draw the incident wavefront AB slanting down so that A touches MN. Mark the angle i between AB and MN.
Draw two incident rays at right angles to AB: one reaching A, and one passing through B to meet MN at C. Write BC = vτ.
With A as centre, draw an arc of radius vτ above MN.
From C, draw a tangent CE to the arc; label it reflected wavefront. Mark the angle r between CE and MN.
Draw reflected rays at right angles to CE, and dashed normals at A and C.

What happens to a plane wavefront in a prism, a lens and a mirror

Device	Emerging wavefront	Why
Thin prism	Plane, but tilted towards the base	Light near the thick base spends longer in glass, where it is slower, so that part lags behind
Convex lens	Converging spherical, meeting at the focus	The middle part passes through the thickest glass and is delayed the most
Concave mirror	Converging spherical, meeting at the focus	The edges of the wavefront reach the curved mirror first and turn back earlier

In real life

A DTH dish antenna on Indian rooftops is a curved reflector. It turns the plane wavefronts from a far-off satellite into converging wavefronts that meet at the receiver placed at its focus.

Exam angle

3 marks: “Use Huygens' principle to verify the law of reflection at a plane surface. Draw a labelled diagram.” Wavefront AB at angle i; BC = vτ; wavelet from A of radius AE = vτ; tangent CE is the reflected wavefront; triangles ABC and AEC congruent (RHS), so i = r; diagram with normals, i and r marked.

2 marks: “Draw the reflected wavefront when a plane wavefront falls on a concave mirror. Where does it go?” A converging spherical wavefront that shrinks to the focus F of the mirror. Diagram: parallel straight lines coming in, arcs centred on F going out.

1 marks: “A plane wavefront passes through a thin prism. What is the shape of the emerging wavefront?” Still plane, but tilted towards the base of the prism.

Remember it

Same medium, same speed, same radius: BC = AE = vτ, so the triangles match and i = r.