MHT-CET Physics · Wave Optics

Young's Double Slit: Fringe Width, Fringe Positions and Shifts

In Young's experiment the bright fringes sit at y = nλD/d and the dark ones halfway between, so the fringe width is β = λD/d; a thin sheet over one slit shifts the whole pattern towards that slit by (μ − 1)tD/d.

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Why this matters

39 PYQs, eight HARD — the largest page in the chapter. Three shapes: how the fringe width responds to λ, D, d or a medium, where a given bright or dark fringe lies (and when fringes of two wavelengths coincide, or a dark fringe falls opposite a slit), and the shift a transparent sheet causes.

Concept 1 of 3: Fringe Width

Fringes are spaced by the distance over which the path difference grows by one wavelength. A longer wavelength or a farther screen spreads them out; wider-set slits crowd them in. In water the wavelength shrinks by μ, and so does the fringe width.

Definition

$\beta = \dfrac{\lambda D}{d}$ ; angular fringe width $\dfrac{\lambda}{d}$ .
$d \times 10$ , $D \times \dfrac{1}{2}$ ⇒ $\beta \times \dfrac{1}{20}$ . Same $\beta$ with $d$ doubled needs $D$ doubled.
In a medium: $\beta' = \dfrac{\beta}{\mu}$ . Violet (short $\lambda$ ) in place of sodium light: narrower fringes.
Fixed region of screen: number of fringes $\propto \dfrac{1}{\lambda}$ (18 at 600 nm ⇒ 27 at 400 nm).
$\beta$ against $D$ is a straight line of slope $\dfrac{\lambda}{d}$ , so $\lambda = \text{slope} \times d$ .

Fringe width

\beta = \frac{\lambda D}{d}

Worked example

\lambda = 500

nm,

D = 1

d = 0.5

mm. Fringe width in air, and with the whole apparatus in water (

\mu = \frac{4}{3}

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 16th May Shift 2 · Q41Easy

Example 1 · Wave Optics · Young's Double Slit — Fringe Width, Positions and Shifts

In a double slit experiment, the distance between slits is increased 10 times, whereas their distance from screen is halved. The fringe width

Multiplying the medium's μ

In water the WAVELENGTH is divided by μ, so the fringe width is divided by μ too. Multiplying gives wider fringes, the wrong way round.

Drill 15 more on fringe width

Concept 2 of 3: Positions of Bright and Dark Fringes

Count from the central bright fringe: the nth bright fringe is n fringe widths out, the nth dark fringe (n − ½). For two points on opposite sides, add the distances; on the same side, subtract. A point opposite one slit is d/2 from the centre, which fixes which fringe can sit there.

Definition

Bright: $y_n = \dfrac{n\lambda D}{d} = n\beta$ . Dark: $y_n = \dfrac{(2n - 1)\lambda D}{2d} = \left(n - \dfrac{1}{2}\right)\beta$ .
13th bright and 4th dark on the same side: $13\beta - 3.5\beta = 9.5\beta$ . 6th dark and 4th bright on opposite sides: $5.5\beta + 4\beta = 9.5\beta$ .
Coincidence: $n$ th bright of $\lambda_1$ on $m$ th dark of $\lambda_2$ : $n\lambda_1 = \left(m - \dfrac{1}{2}\right)\lambda_2$ .
Dark fringe opposite a slit ( $y = \dfrac{d}{2}$ ): $\lambda = \dfrac{d^2}{(2n - 1)D}$ — first dark $\dfrac{d^2}{D}$ , second $\dfrac{d^2}{3D}$ , third $\dfrac{d^2}{5D}$ .

Fringe positions

y_{\text{bright}} = n\beta, \qquad y_{\text{dark}} = \left(n - \tfrac{1}{2}\right)\beta

Worked example

The fringe width is 0.4 mm. How far apart are the 5th bright and the 3rd dark fringe on the same side?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 9th May Shift 2 · Q40Moderate

Example 2 · Wave Optics · Young's Double Slit — Fringe Width, Positions and Shifts

In Young's double slit experiment, the fringe width is 2 mm. The separation between the 13th bright fringe and the 4th dark fringe from the centre of the screen on the same side will be

Placing the nth dark fringe at nβ

The first dark fringe is only HALF a fringe width out. The 4th dark is at

3.5\beta

, not

4\beta

— and the difference is always one of the options.

Drill 16 more on positions of bright and dark fringes

Concept 3 of 3: A Transparent Sheet Over One Slit

A sheet of thickness t and index μ makes light through it travel (μ − 1)t of extra optical path. The point of zero path difference — the central bright fringe — moves towards the covered slit until the geometry makes up that extra path. The fringe width does not change.

Definition

Extra optical path $(\mu - 1)t$ ; shift $= \dfrac{(\mu - 1)tD}{d}$ , i.e. $\dfrac{(\mu - 1)t}{\lambda}$ fringes.
The pattern shifts TOWARDS the covered slit; fringe width and number are unchanged.
Same thickness, different sheets: shift $\propto (\mu - 1)$ .

Fringe shift

\Delta y = \frac{(\mu - 1)tD}{d}, \qquad N = \frac{(\mu - 1)t}{\lambda}

Worked example

A sheet 5 μm thick with

\mu = 1.6

covers one slit;

\lambda = 600

nm. By how many fringes does the pattern shift?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 14th May Shift 2 · Q34Moderate

Example 3 · Wave Optics · Young's Double Slit — Fringe Width, Positions and Shifts

On replacing a thin film of mica of thickness

12 \times 10^{-5}\,\text{cm}

in the path of one of the interfering beams in Young's double slit experiment, the fringe pattern shifts by one fringe width. If

\lambda = 6 \times 10^{-5}\,\text{cm}

, the refractive index of mica is

Using μt instead of (μ − 1)t

The sheet REPLACES a thickness t of air, so the extra path is

(\mu - 1)t

. Using

\mu t

makes the shift about three times too big.

Drill 5 more on a transparent sheet over one slit

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Fringe Width
Fringe width
$\beta = \frac{\lambda D}{d}$
Positions of Bright and Dark Fringes
Fringe positions
$y_{\text{bright}} = n\beta, \qquad y_{\text{dark}} = \left(n - \tfrac{1}{2}\right)\beta$
A Transparent Sheet Over One Slit
Fringe shift
$\Delta y = \frac{(\mu - 1)tD}{d}, \qquad N = \frac{(\mu - 1)t}{\lambda}$

Watch out for (3)

Multiplying the medium's μ
→ Fringe Width
Placing the nth dark fringe at nβ
→ Positions of Bright and Dark Fringes
Using μt instead of (μ − 1)t
→ A Transparent Sheet Over One Slit

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Related notes

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