Exercises: Young's slits and diffraction gratings

A diffraction grating has $500$ lines per millimetre.

1. a) Calculate the slit spacing $d$ in metres.
2. b) State the grating equation $n\lambda = d\sin\theta$ and name each symbol.
3. c) Calculate the angle of the first-order maximum for light of wavelength $600\ \text{nm}$.

In a Young's double-slit experiment in a school laboratory in Tullamore, the slit separation is $0.30\ \text{mm}$ and the screen is $2.0\ \text{m}$ from the slits. The fringe spacing is given by $\Delta y=\dfrac{\lambda D}{d}$.

1. a) Calculate the fringe spacing for light of wavelength $550\ \text{nm}$.
2. b) State what happens to the fringe spacing if the slit separation is doubled.
3. c) State what is observed at the centre of the pattern and explain why.

A grating with slit spacing $2.0\ \mu\text{m}$ is illuminated with monochromatic light of wavelength $500\ \text{nm}$.

1. a) Calculate the angle of the first-order maximum.
2. b) Calculate the angle of the second-order maximum.
3. c) Calculate the largest order of maximum that can be observed, and explain why.

A student shines a red laser through a diffraction grating with $300$ lines per millimetre. The first-order maximum appears at an angle of $11.0^\circ$ to the straight-through direction.

1. a) Calculate the slit spacing.
2. b) Calculate the wavelength of the laser light.
3. c) Calculate the angle of the second-order maximum.
4. d) Calculate the number of maxima visible on each side of the central maximum.

In an experiment to determine the wavelength of light using a double slit of separation $0.25\ \text{mm}$, a student measures the distance across $6$ fringe spacings on a screen $1.80\ \text{m}$ away as $24.0\ \text{mm}$.

1. a) Calculate the fringe spacing.
2. b) Calculate the wavelength of the light in nanometres.
3. c) Suggest why measuring across several fringes gives a more reliable result than measuring a single fringe.
4. d) Name the colour of light that has approximately this wavelength.

A student measures the angle $\theta$ of several orders $n$ of a laser light through a grating.

Order $n$	$\theta$ / degrees
1	18.5
2	39.4
3	73.6

The grating has $1,000$ lines per millimetre.

1. a) Calculate the wavelength from each measurement.
2. b) Calculate the mean wavelength.
3. c) Explain how a graph of $\sin\theta$ against $n$ could be used to find the wavelength, and state what the gradient would be.

White light, with wavelengths from $400\ \text{nm}$ to $700\ \text{nm}$, is incident normally on a grating with $400$ lines per millimetre.

1. a) Calculate the angular positions of the first-order maximum for violet light ($400\ \text{nm}$) and red light ($700\ \text{nm}$).
2. b) Calculate the angular width of the first-order spectrum.
3. c) Show whether the second-order and third-order spectra overlap, by comparing the angle for red in second order with the angle for violet in third order.
4. d) State what is seen at the central maximum and explain why.

A sodium lamp emits two close wavelengths of $589.0\ \text{nm}$ and $589.6\ \text{nm}$. A grating has $600$ lines per millimetre and a screen is placed $2.00\ \text{m}$ from the grating.

1. a) Calculate the angle of the second-order maximum for each wavelength.
2. b) Calculate the separation of the two second-order lines on the screen, using small angle approximations where suitable and stating where you have used them.
3. c) Explain why a grating with more lines per millimetre gives better separation of the two lines.
4. d) Explain why a grating gives sharper and brighter maxima than a double slit for the same wavelength.

A student uses a double slit and a grating to compare their results with the same green laser. With a double slit of separation $0.20\ \text{mm}$ and a screen at $3.00\ \text{m}$, the fringe spacing is $8.1\ \text{mm}$. With a grating of $800$ lines per millimetre, the first-order maximum appears at an angle of $24.9^\circ$.

1. a) Calculate the wavelength from the double-slit data.
2. b) Calculate the wavelength from the grating data.
3. c) Calculate the percentage difference between the two values and comment on whether they agree within experimental error.
4. d) Identify the main source of uncertainty in each experiment, and suggest an improvement in each case.