Questions on interference and Young's double slit

Question 1

State the condition, in terms of path difference, for constructive and for destructive interference.

Accepted Answer

Constructive interference: path difference = nλ (1). Destructive interference: path difference = (n + ½)λ, where n is an integer (1).

Question 2

Explain what is meant by two sources being coherent.

Accepted Answer

Coherent sources have the same frequency (and wavelength) (1) and keep a constant phase difference between them over time (1).

Question 3

State the double-slit fringe-spacing equation and name each quantity in it.

Accepted Answer

w = λD/s (1), where w is the fringe spacing, λ the wavelength, D the distance from the slits to the screen, and s the slit separation (1).

Question 4

The fringe pattern produced in a double-slit experiment is too cramped to measure accurately. State two changes to the arrangement, without changing the light source, that would increase the fringe spacing.

Accepted Answer

Move the screen further from the slits, increasing D (1). Use a slit pair with a smaller separation s (1).

Question 5

In one version of Young's experiment, a single narrow slit is placed between a lamp and the double slits. Explain why the light leaving the two slits is coherent.

Accepted Answer

Both slits are illuminated by the same wavefront from the one source (single slit), so the light at the two slits has the same frequency (1) and a constant phase difference (1).

Question 6

A point on the screen in a double-slit experiment is 2.5 wavelengths further from one slit than from the other. State and explain what is seen at this point.

Accepted Answer

A dark fringe (1). The path difference is (n + ½)λ, so the two waves arrive in antiphase and interfere destructively (1).

Question 7

In a double-slit experiment, light of wavelength 600 nm illuminates slits 0.50 mm apart, and the screen is 1.5 m away. Calculate the fringe spacing.

Accepted Answer

w = λD/s (1) = (600 × 10⁻⁹ × 1.5)/(0.50 × 10⁻³) (1) w = 1.8 mm (1)

Question 8

A double-slit pattern has a fringe spacing of 2.4 mm on a screen 2.0 m from slits 0.40 mm apart. Calculate the wavelength of the light.

Accepted Answer

λ = ws/D (1) = (2.4 × 10⁻³ × 0.40 × 10⁻³)/2.0 (1) λ = 480 nm (1)

Question 9

Light of wavelength 650 nm produces fringes 1.3 mm apart on a screen 1.2 m from the slits. Calculate the slit separation.

Accepted Answer

s = λD/w (1) = (650 × 10⁻⁹ × 1.2)/(1.3 × 10⁻³) (1) s = 0.60 mm (1)

Question 10

In a double-slit experiment using 600 nm light, state the path difference at the second bright fringe from the centre, giving your answer in micrometres.

Accepted Answer

Second bright fringe: path difference = 2λ = 2 × 600 nm (1) = 1200 nm = 1.2 μm (1)

Question 11

Two loudspeakers connected to the same signal generator emit sound of frequency 850 Hz in phase. The speed of sound in air is 340 m s⁻¹. A listener stands at a point where the distances to the two speakers differ by 0.60 m. Deduce what the listener hears at this point.

Accepted Answer

λ = v/f = 340/850 = 0.40 m (1). Path difference = 0.60/0.40 = 1.5λ (1). A half-integer number of wavelengths means the waves arrive in antiphase, so the interference is destructive and the listener hears a minimum (a quiet point) (1).

Question 12

In a double-slit experiment, a student measures the distance across 8 fringe spacings as 21.6 mm to reduce the uncertainty. The slit separation is 0.40 mm and the screen is 1.8 m from the slits. Calculate the wavelength of the light.

Accepted Answer

w = 21.6 mm/8 = 2.7 mm (1). λ = ws/D = (2.7 × 10⁻³ × 0.40 × 10⁻³)/1.8 (1) = 6.0 × 10⁻⁷ m = 600 nm (1).

Question 13

A teacher wants double-slit fringes at least 4.0 mm apart so a class can see them. The laser wavelength is 633 nm and the slit separation is 0.30 mm. Calculate the minimum distance needed between the slits and the screen.

Accepted Answer

Rearranging w = λD/s: D = ws/λ (1) = (4.0 × 10⁻³ × 0.30 × 10⁻³)/(633 × 10⁻⁹) (1) = 1.9 m (1).

Question 14

In a double-slit experiment, 500 nm light and a screen distance of 2.0 m give fringes with slits 0.25 mm apart. Calculate the fringe spacing, then state and calculate the new spacing if the slit separation is halved.

Accepted Answer

w = λD/s = (500 × 10⁻⁹ × 2.0)/(0.25 × 10⁻³) (1) w = 4.0 mm (1) Halving s doubles w (w ∝ 1/s) (1) new w = 8.0 mm (1)

Question 15

Explain why coherent, monochromatic light is needed to see clear double-slit fringes, and describe the central fringe seen when white light is used instead.

Accepted Answer

Coherence keeps the phase difference between the two beams constant, so the bright and dark fringes stay in fixed positions (1); a single wavelength gives one fringe spacing so the fringes are sharp (1). With white light the central fringe is white (all wavelengths meet in phase there) (1), with coloured fringes either side that quickly overlap and wash out (1).

Question 16

Explain why a stable interference pattern is not seen when light from two separate lamps overlaps.

Accepted Answer

The two lamps are incoherent (1): they emit light with a randomly and rapidly changing phase difference (1). The positions of constructive and destructive interference shift far too quickly to be seen, so the pattern averages out to uniform brightness (1).

Question 17

Green light of wavelength 546 nm from a filtered mercury lamp passes through double slits of separation 0.45 mm. The screen is 2.2 m from the slits. Show that the fringe spacing is about 2.7 mm. Go on to determine the number of bright fringes seen within the central 2.0 cm of the pattern, centred on the middle bright fringe.

Accepted Answer

w = λD/s = (546 × 10⁻⁹ × 2.2)/(0.45 × 10⁻³) (1) = 2.67 mm ≈ 2.7 mm (1). Fringes either side of the centre: 10 mm/2.67 mm = 3.7, so 3 whole spacings fit each side (1). Number of bright fringes = 2 × 3 + 1 = 7 (1).

Question 18

A student has a laser of wavelength 650 nm, and the bench allows a maximum slit-to-screen distance of 1.4 m. Two slit pairs are available.
Slit pair P: separation 0.30 mm, slit width 0.10 mm.
Slit pair Q: separation 0.60 mm, slit width 0.10 mm.
The student needs a fringe spacing of at least 2.0 mm to measure it reliably with a ruler. Deduce which slit pair the student should use.

Accepted Answer

Use of w = λD/s with D = 1.4 m (1). Pair P: w = (650 × 10⁻⁹ × 1.4)/(0.30 × 10⁻³) = 3.0 mm (1). Pair Q: w = 1.5 mm (1). Only pair P gives a spacing of at least 2.0 mm, so the student should use P; the slit width is not needed in the comparison (1).

Question 19

A student is given a laser of unknown wavelength, a double-slit slide with the slit separation 0.50 mm printed on it, a ruler, a tape measure and a screen. Describe how the student should determine the wavelength of the laser light. Include the measurements taken, how the wavelength is calculated from them, two ways of keeping the uncertainty small, and one relevant safety precaution.

Accepted Answer

Shine the laser normally through the double slits so a fringe pattern forms on the screen a few metres away, in a darkened room (1). Measure the slit-to-screen distance D with the tape measure (1). Measure the distance across many fringe spacings with the ruler and divide by the number of spacings to find w (1). Calculate the wavelength from λ = ws/D (1). Uncertainty: use a large D so the fringes are widely spaced, measure across as many fringes as possible, and repeat the measurements and average (any two) (1). Safety: do not look into the beam or let reflections from shiny surfaces reach anyone's eyes (1).