A full lesson: a cartoon movie at a wedding hall, then superposition in a ripple tank, path and phase difference, the intensity formula built from two waves, why ordinary lamps never show fringes, Young's clever trick, NEET shortcuts, problems to solve yourself, and 12 MCQs.
Two speakers playing the same steady tone make fixed loud and soft spots in the hall. Where the two paths differ by half a wavelength, the waves cancel. Light does the same, but only if both waves come from one source.
Before a wedding sangeet, Rohan uncle tests the sound system: both speakers on the stage play one steady "hmmmm" tone. Priya walks across the empty hall and notices something strange: in some places the tone is loud, a few steps later it almost disappears, then it is loud again. The speakers did not change. Only her position did.
At each spot two waves arrive, one from each speaker. If crest meets crest they add up; if crest meets trough they cancel. Which one happens depends on how much farther one speaker is than the other: the path difference. This is interference, and light does exactly the same thing, as Thomas Young showed in 1801.
Put two phone speakers (or two earphones) half a metre apart playing the same steady tone from one tone-generator app with a split cable. Close one ear and move your head slowly sideways. Do you hear loud and soft places? Now play two different songs from two phones. What changes?
With one tone fed to both, you hear fixed loud and soft spots: the sources are coherent. Two different songs (or two separate phones) have no fixed phase relation, so there are no steady quiet spots: the sounds just add up.
"Two waves, one place. Crest on crest: louder. Crest on trough: silence. The secret is a steady phase difference."
Two needles dip into water, moving up and down exactly alike. This picture is computed live by adding the two circular waves at every point. Change the separation, switch between the instant ripples and the average brightness, and drag the probe P.
The resultant displacement at any point is the (vector) sum of the displacements of the individual waves.
Not "equal brightness" or "same colour". Only a phase difference that does not change.
With coherent sources the pattern stays fixed. More separation d gives more lines, packed closer together.
"Look at the average view. With coherent sources, the bright and dark lines stay put. Switch to incoherent and they smear into a flat grey: no interference pattern."
Two in-phase sources S₁ and S₂ send waves of equal amplitude a to a point P. One path is longer by Δ = S₂P − S₁P. Slide Δ and watch wave 1 (blue), wave 2 (purple) and their sum (green).
One whole wavelength of extra path is one whole cycle, 2π, of phase.
Crest meets crest. Resultant amplitude 2a, intensity 4I₀.
Crest meets trough. For equal amplitudes the resultant is zero.
"Whole number of wavelengths: bright. Half-odd number: dark. On the perpendicular bisector Δ = 0, so that line is always bright."
"What if S₂ vibrates exactly opposite to S₁ (phase difference π at the source)? Is the bisector still bright?"
No, it becomes dark. Add the source phase difference to (2π/λ)Δ. With an extra π, every bright place becomes dark and every dark place bright.
Between "fully bright" and "fully dark" there is everything in between. Add the two waves as arrows (phasors), and read the intensity on the graph. Drag the marker on the graph or use the sliders.
Equal amplitude a, phase difference φ at the point P.
The resultant is a wave of the same frequency with amplitude 2a cos(φ/2).
I₀ ∝ a² is the intensity of one wave alone. Maximum 4I₀ at φ = 0, 2π, 4π…; zero at φ = π, 3π…
I_max = (√I₁ + √I₂)², I_min = (√I₁ − √I₂)². With I₁ = I₂ = I₀ this is the formula above.
"Check any formula with three numbers: φ = 0 gives 4I₀, φ = π gives 0, and the average over all φ is 2I₀. Energy is not created or lost, only moved from dark places to bright places."
A real lamp changes its phase abruptly about every 10⁻¹⁰ s. Below, the top strip shows the fringe pattern at one instant; the graph shows what a detector records on average. Switch between coherent and incoherent sources.
Fixed φ at each point: steady bright (4I₀) and dark (0) fringes.
φ changes randomly and rapidly, so only the average is seen: uniform 2I₀, no fringes.
For incoherent sources add intensities. For coherent sources add amplitudes first, then square.
"Two candles, two bulbs, two sodium lamps: incoherent. Their intensities add, 2I₀ everywhere. That is why your room has no dark stripes on the wall."
Each lamp below changes its phase at random moments (the colour change on its waves). Compare two separate lamps with Young's arrangement: one pinhole S lighting two close pinholes S₁ and S₂.
Each atom emits independently. The phase difference of two lamps changes billions of times a second.
A phase jump at S reaches S₁ and S₂ at the same moment (equal paths), so φ₁ − φ₂ stays fixed.
Light reflected from the top and bottom of a thin film comes from the same wave: coherent. The colours are interference.
"Think of S as the parent and S₁, S₂ as twins. Whatever the parent does, both twins do at the same moment. That is coherence."
λ/4 → π/2, λ/2 → π, λ → 2π, 1.5λ → 3π.
φ = 60°: 3I₀; 90°: 2I₀; 120°: I₀; 180°: 0.
I and 4I: 9 : 1. I and 9I: 4 : 1.
a₁ : a₂ = 2 : 1 gives 9 : 1. Work backwards with square roots.
Average of a coherent pattern is also 2I₀: energy is conserved, only redistributed.
Unequal sources: dark fringes are not fully dark (I_min ≠ 0).
"For any interference number, ask two questions: coherent or not? Equal intensities or not? Then pick the formula. Half the NEET questions end right there."
Cover the solution, write the formula, and solve on paper. Open the hint only if you are stuck for more than a minute.
Two coherent sources each give intensity I₀ at a point. The phase difference there is π/3. Find the resultant intensity and the path difference (in terms of λ).
I = 4I₀ cos²(φ/2); Δ = (λ/2π) φ.
Light of wavelength 600 nm from two in-phase coherent sources reaches a point with a path difference of 1.5 μm. Is the point bright or dark? What is the phase difference?
Divide Δ by λ. Whole number: bright. Half-odd: dark.
Two coherent sources have intensities I and 4I. Find the maximum and minimum intensity in the interference pattern and their ratio.
I_max = (√I₁ + √I₂)², I_min = (√I₁ − √I₂)².
Two identical bulbs each give intensity I₀ on a wall. What is the intensity on the wall? If instead two coherent sources of the same strength are used, what are the brightest, darkest and average intensities?
Incoherent: add intensities. Coherent: 4I₀ cos²(φ/2), average of cos² is ½.
In an interference pattern I_max : I_min = 25 : 9. Find the ratio of the amplitudes and of the intensities of the two sources.
Take square roots: (a₁ + a₂)/(a₁ − a₂) = 5/3.
I₀ is the intensity of one wave alone. Write the formula first, then click.
Score: 0 / 12
| If the question says… | Write this first | The trap |
|---|---|---|
| "coherent sources" defined | constant phase difference in time | Choosing "same amplitude" or "same intensity" |
| "path difference x, find phase" | φ = 2πx/λ | Forgetting the 2π, or using λ/x |
| "intensity at phase difference φ" | I = 4I₀ cos²(φ/2) | Writing cos²φ instead of cos²(φ/2) |
| "maximum intensity" with I₀ each | 4I₀ (not 2I₀) | Adding intensities for coherent waves |
| "two independent sources / two bulbs" | incoherent: I = I₁ + I₂, no fringes | Answering 4I₀ and 0 |
| "ratio of max to min intensity" | ((√I₁ + √I₂)/(√I₁ − √I₂))² | Using the intensity ratio directly without square roots |
| "amplitude ratio a₁ : a₂" | I_max/I_min = ((a₁ + a₂)/(a₁ − a₂))² | Taking square roots of amplitudes |
| "energy in interference" | conserved, redistributed | "Energy is destroyed at dark fringes" |
Lesson 4 puts numbers on Young's arrangement: the path difference x d/D on a screen, the positions of bright and dark fringes, and the fringe width β = λD/d.
Notation: I₀ = intensity of one wave alone (∝ a²), φ = phase difference, Δ = path difference. NCERT §10.4 treats equal amplitudes; the unequal-intensity formulas are standard NEET extensions.
Follow the four steps: understand it, remember it, try it, then prove it with past papers.