Interference

SYLLABUS

Phenomenon of Interferences: Coherent sources, Young’s double slit experiment.

What is This Chapter About?

Shine light through two closely spaced slits and look at the pattern formed on a screen behind them. Instead of seeing two bright lines corresponding to the two slits, you see a beautiful alternating pattern of bright and dark bands — interference fringes. This experiment, first performed by Thomas Young in 1801, was decisive proof that light behaves as a wave. No particle model could explain the appearance of dark bands where light from two sources arrives simultaneously.

Interference is the phenomenon that occurs when two or more waves from coherent sources superpose. It is one of the most elegant and fundamental demonstrations of wave behaviour in nature, and it forms the conceptual heart of wave optics. Chapter 11 of NEB Grade 12 Physics carries 2 teaching hours as a focused and mathematically precise introduction to this phenomenon, laying the groundwork for understanding optical instruments, thin film colours, and anti-reflection coatings.

Key Concepts You Will Learn

Superposition of Waves

The foundation of interference is the principle of superposition: when two or more waves meet at a point, the resultant displacement at that point is the algebraic sum of the individual displacements. If two waves arrive in phase (crest meets crest), they add constructively — constructive interference — producing a bright fringe with amplitude 2A and intensity [latex]4I_0[/latex]. If they arrive exactly out of phase (crest meets trough), they cancel — destructive interference — producing a dark fringe with zero intensity.

Coherent Sources

For a stable, observable interference pattern, the two sources must be coherent — they must maintain a constant phase difference with respect to each other. If the phase difference fluctuates randomly, the bright and dark fringes shift continuously and wash out into a uniform illumination. In practice, true coherence requires that both waves originate from the same source — which is why Young used a single light source illuminating two narrow slits rather than two separate sources.

In modern optics, lasers provide highly coherent light and make interference experiments much easier to perform than with conventional light sources. The temporal coherence of a laser is far superior to that of a lamp or sunlight.

Young’s Double Slit Experiment

Young’s double slit experiment is the classic demonstration of light interference. A single monochromatic light source illuminates a barrier with two narrow, closely spaced slits separated by distance d. Light from both slits diffracts and overlaps on a screen at distance D from the slits, creating a pattern of alternating bright and dark fringes.

The condition for constructive interference (bright fringe) is that the path difference from the two slits to the point on the screen must be a whole number of wavelengths:

[latex]\Delta x = n\lambda[/latex], where n = 0, ±1, ±2, …

The condition for destructive interference (dark fringe) is that the path difference must be a half-integer number of wavelengths:

[latex]\Delta x = (2n-1)\frac{\lambda}{2}[/latex], where n = 1, 2, 3, …

For a point P at height y above the central axis on the screen, the path difference is approximately [latex]\Delta x = \frac{yd}{D}[/latex] for small angles. Applying the condition for bright fringes gives the position of the nth order maximum:

[latex]y_n = \frac{n\lambda D}{d}[/latex]

The fringe width — the distance between consecutive bright (or dark) fringes — is constant across the pattern:

[latex]\beta = \frac{\lambda D}{d}[/latex]

This elegant result means that fringe width increases with wavelength (red light gives wider fringes than blue), increases with screen distance, and decreases with slit separation. Measuring the fringe width in a Young’s double slit experiment allows the wavelength of light to be determined precisely — one of the earliest methods of measuring the wavelength of visible light.

Intensity Distribution

The intensity at any point in the interference pattern is: [latex]I = 4I_0\cos^2\frac{\delta}{2}[/latex], where [latex]\delta[/latex] is the phase difference between the two waves and [latex]I_0[/latex] is the intensity from each slit alone. This smooth cosine-squared variation — maximum at the bright fringes, zero at the dark fringes — is the hallmark of two-beam interference and distinguishes it from the more complex patterns produced by multiple slits or diffraction gratings.

Why This Chapter Matters for NEB Exams

Interference is examined consistently in NEB Grade 12 Physics. Young’s double slit experiment — explaining the setup, deriving the fringe width formula, and applying it numerically — is one of the most classic long-answer questions in the entire course. Conditions for constructive and destructive interference, the definition of coherent sources, and the intensity distribution formula are standard short and medium-answer questions. Numerical problems involving fringe width, wavelength, slit separation, and screen distance appear in virtually every NEB board paper.

Important Topics to Focus On

• Principle of superposition — constructive and destructive interference
• Definition and importance of coherent sources
• Why two independent light sources cannot produce stable interference
• Young’s double slit experiment — setup and diagram
• Path difference conditions: [latex]\Delta x = n\lambda[/latex] for bright, [latex]\Delta x = (2n-1)\frac{\lambda}{2}[/latex] for dark
• Derivation of position of nth order maximum: [latex]y_n = \frac{n\lambda D}{d}[/latex]
• Derivation of fringe width: [latex]\beta = \frac{\lambda D}{d}[/latex]
• Effect of changing [latex]\lambda[/latex], D, and d on fringe width
• Intensity distribution: [latex]I = 4I_0\cos^2\frac{\delta}{2}[/latex]
• Using fringe width to calculate wavelength of light
• Numerical problems on all of the above

Connection to Other Chapters

Interference builds directly on Chapter 10 (Nature and Propagation of Light) — Huygens’ Principle explains why light from each slit spreads out (diffracts) and overlaps to create the interference pattern. Chapter 12 (Diffraction) extends these ideas to single slit and grating patterns, where interference between many sources creates more complex intensity distributions. Chapter 13 (Polarization) uses the transverse wave nature of light — demonstrated by interference experiments — to explain polarization. The wavelength measurement from Young’s experiment connects to Chapter 21 (Photons), where wavelength relates to photon energy through [latex]E = hf[/latex]. Interference also appears in Chapter 8 (Waves in Pipes and Strings) — stationary waves are a special case of interference between waves travelling in opposite directions.