The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Chapter 36 — Diffraction

Problem sheet · 14 points · 3 extra credit

Chapter 36 — Diffraction

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (2 pts ••) Monochromatic light of wavelength $\lambda=633\,\text{nm}$ from a helium–neon laser passes through a single slit of width $a=0.12\,\text{mm}$ and forms a diffraction pattern on a screen $L=2.5\,\text{m}$ away. The angles are small enough that $\sin\theta\approx\tan\theta$.

   (a) Find the angle to the first dark fringe ($m=1$).

   (b) Find the distance on the screen from the center of the pattern to the first dark fringe.

   (c) Find the full width of the central bright band (the distance between the first dark fringes on either side).

2. 2. (2 pts ••) A single slit of width $a=0.15\,\text{mm}$ is illuminated by light of wavelength $\lambda=600\,\text{nm}$. The intensity as a function of the diffraction angle is

   $$I=I_0\left[\frac{\sin(\beta/2)}{\beta/2}\right]^2,\qquad \beta=\frac{2\pi a}{\lambda}\sin\theta .$$

   (a) Find the angle at which $\beta/2=\pi/2$ (i.e. halfway, in the phase variable, to the first minimum).

   (b) Find the intensity at that angle as a fraction of the central intensity $I_0$.

   (c) The first secondary maximum occurs near $\beta/2=1.43\pi$. Find its intensity as a fraction of $I_0$, and comment on how much dimmer it is than the central peak.

3. 3. (2 pts ••) A diffraction grating has $600$ lines per millimetre and is illuminated at normal incidence by sodium light, $\lambda=589\,\text{nm}$.

   (a) Find the slit spacing $d$.

   (b) Find the diffraction angles of the first- and second-order bright fringes.

   (c) What is the highest order that can be observed with this grating and wavelength? Explain.

4. 4. (2 pts ••) The sodium doublet consists of two closely spaced lines at $\lambda_1=589.00\,\text{nm}$ and $\lambda_2=589.59\,\text{nm}$. The resolving power of a grating is $R=Nm$, where $N$ is the number of illuminated slits and $m$ is the order.

   (a) What resolving power is needed to just separate the two lines?

   (b) Working in second order ($m=2$), how many slits must be illuminated?

   (c) If the grating has $600$ lines per millimetre, what width of the grating must the beam cover to achieve this?

5. 5. (2 pts ••) A camera lens and a telescope are limited by the diffraction of a circular aperture; the Rayleigh criterion gives the smallest resolvable angular separation as $\theta=1.22\,\lambda/D$. Take $\lambda=550\,\text{nm}$ throughout.

   (a) A space telescope has a primary mirror of diameter $D=2.4\,\text{m}$. Find its diffraction-limited angular resolution, in radians and in arcseconds.

   (b) Aimed at the Moon ($3.84\times10^{8}\,\text{m}$ away), what is the smallest surface separation it could resolve?

   (c) The dark-adapted human eye has a pupil of diameter $D=3.0\,\text{mm}$. How far away can two headlights separated by $1.5\,\text{m}$ be and still be resolved as two sources?

6. 6. (2 pts ••) X-rays are directed at a crystal of table salt (NaCl), whose atomic planes are separated by $d=0.282\,\text{nm}$. A strong first-order reflection ($m=1$) is observed when the beam makes a grazing angle of $\theta=15.9^\circ$ with the planes. Bragg's law is $2d\sin\theta=m\lambda$.

   (a) Find the wavelength of the X-rays.

   (b) Find the grazing angle at which the second-order reflection ($m=2$) appears.

   (c) What is the highest order that can be observed with these X-rays and this spacing?

7. 7. (2 pts ••) A double-slit apparatus has slits of width $a=0.050\,\text{mm}$ with their centers separated by $d=0.25\,\text{mm}$, illuminated by light of wavelength $\lambda=550\,\text{nm}$. The observed pattern is the two-slit interference fringes modulated by the single-slit diffraction envelope.

   (a) At what interference orders $m$ are bright fringes missing because they fall on a zero of the single-slit envelope? Identify the lowest such order.

   (b) How many bright interference fringes appear within the central diffraction maximum (counting the central fringe and those on both sides, but excluding any that land exactly on the first diffraction minimum)?

8. Extra Credit (3 pts •••)

   Derive the single-slit intensity pattern from scratch using phasor (amplitude) integration.

   Consider a slit of width $a$ illuminated by a plane wave. Divide the slit into infinitesimal strips of width $dy$ at position $y$ (with $-a/2\le y\le a/2$). Each strip radiates a wavelet toward a distant screen at angle $\theta$; relative to the strip at $y=0$, the wavelet from the strip at $y$ has an extra path length $y\sin\theta$, hence a phase $\phi(y)=ky\sin\theta$ with $k=2\pi/\lambda$. Each strip contributes an amplitude $dE_0=(E_0/a)\,dy$, where $E_0$ is the total amplitude that would arrive with all strips in phase.

   (a) Write the total (complex) field at the screen as the integral $E=\displaystyle\int_{-a/2}^{a/2}\frac{E_0}{a}\,e^{i\,ky\sin\theta}\,dy$, and carry out the integration.

   (b) Show that the magnitude reduces to

   $$|E|=E_0\,\frac{\sin(\beta/2)}{\beta/2},\qquad \beta=\frac{2\pi a}{\lambda}\sin\theta ,$$

   and hence that the intensity (proportional to $|E|^2$) is $I=I_0\left[\dfrac{\sin(\beta/2)}{\beta/2}\right]^2$.

   (c) From this result, show that the minima occur at $a\sin\theta=m\lambda$ for nonzero integers $m$, and use a limit to verify that $I\to I_0$ as $\theta\to0$.