8 min readConcept sheet1-mark MCQ, 2-mark 'why can two bulbs not give interference', and the conditions for sustained interference as a 2 or 3-mark answer

Coherent sources and sustained interference

Coherent sources keep a constant phase difference. Only they give a steady, sustained interference pattern; two separate bulbs never do.

Coherent sourcesSame frequency and a phase difference that does not change with time

Coherent addition (equal waves)I = 4I₀ cos²(φ/2), changing from 0 to 4I₀: fringes

Incoherent addition (equal waves)I = 2I₀ at every point: no fringes

How to get themSplit light from one source into two parts, as the two slits in Young's experiment do

Coherent vs incoherent sources

CoherentEmit waves of the same frequency with a constant phase difference φ. At each point φ stays fixed, so the intensity there stays fixed at 4I₀ cos²(φ/2). We see steady bright and dark fringes. Example: the two slits of Young's experiment, lit by one source.

IncoherentThe phase difference changes randomly and very fast. At each point, cos²(φ/2) averages to ½, so I = 2I₀ everywhere. The screen looks evenly lit. Example: two separate bulbs or two sodium lamps.

Why two bulbs never give interference

Light from an ordinary lamp comes from a huge number of atoms. Each atom sends out light for a very short time, and the phase of the light changes in a random way. For two separate lamps, the phase difference changes randomly, far faster than our eyes or any detector can follow. We only see the average, which is 2I₀ everywhere. So we make coherent sources by dividing light from one source into two parts. In Young's experiment, the two slits pick up parts of the same wavefront. Any sudden phase change in the source reaches both slits together, so their phase difference stays the same.

Conditions for sustained (steady) interference

The two sources must be coherent: same frequency and constant phase difference. So they must come from one parent source.
The light should be monochromatic (one wavelength). Otherwise, fringes of different colours overlap and blur.
The two waves should have equal or nearly equal amplitudes, so the dark fringes are really dark (good contrast).
The sources (slits) should be narrow, so that each acts as a fine line source.
The slits should be close together and the screen fairly far away, so the fringes are wide enough to see (β = λD/d).

Worked example: coherent and incoherent addition

Problem

Each of two sources alone gives an intensity of 2 units at a point on a screen. Find the intensity at a point where the phase difference is (i) 0 and (ii) π, if the sources are (a) coherent, (b) incoherent.

Working

Here I₀ = 2 units.
(a) Coherent: I = 4I₀ cos²(φ/2).
(i) φ = 0: I = 4 × 2 × cos² 0 = 8 units (bright).
(ii) φ = π: I = 4 × 2 × cos²(π/2) = 0 (dark).
(b) Incoherent: the average of cos²(φ/2) is ½, so I = 2I₀ = 4 units at both points.

Answer

Coherent: 8 units and 0. Incoherent: 4 units at both points, so no fringes.

In real life

The two headlights of a car never make bright and dark bands on the road, because they are independent, incoherent sources. A laser pointer gives highly coherent light, so it easily shows interference in a school laboratory.

Exam angle

2 marks: “Why can two independent sodium lamps not produce a sustained interference pattern?” They are incoherent: their phase difference changes randomly and very fast. The eye sees only the average intensity, 2I₀, everywhere, so no steady fringes form.

2 marks: “Write any two conditions for sustained interference of light.” Any two: coherent sources from one parent source (same frequency, constant phase difference); nearly equal amplitudes; narrow slits close together; monochromatic light.

3 marks: “Distinguish between coherent and incoherent addition of light waves. Give the intensity in each case for two waves of intensity I₀.” Coherent: constant φ; I = 4I₀ cos²(φ/2), varying from 0 to 4I₀; fringes seen. Incoherent: φ changes randomly; average of cos² is ½; I = 2I₀ everywhere; no fringes. Examples: two slits lit by one source; two bulbs.

Remember it

Coherent = one parent, fixed phase. Two bulbs have two parents, so no fringes.