"Dual Nature of Matter and Radiation" — Practice

Question 1

Light of wavelength $350$ nm falls on a metal whose threshold wavelength is $500$ nm. Find the work function, the maximum kinetic energy and the stopping potential.

Accepted Answer

$\phi=\dfrac{1240}{500}=2.48$ eV

$E=\dfrac{1240}{350}=3.54$ eV

$K_{max}=3.54-2.48=1.06$ eV

$V_0=1.06$ V, since $K_{max}$ in electronvolts equals $V_0$ in volts numerically.

Question 2

When light of frequency $
u$ falls on a metal the stopping potential is $V_1$, and at frequency $2
u$ it is $V_2$. Show that $V_2>2V_1$ and find the work function in terms of $V_1$, $V_2$ and $e$.

Accepted Answer

$eV_1=h
u-\phi$ and $eV_2=2h
u-\phi$.

Subtracting: $e\left(V_2-V_1ight)=h
u=eV_1+\phi$

$\phi=e\left(V_2-2V_1ight)$

Since $\phi>0$, it follows that $V_2>2V_1$: doubling the frequency more than doubles the stopping potential, because the work function is subtracted only once.

Question 3

In a photoelectric experiment the intensity of the incident light is doubled at fixed frequency. State what happens to the saturation current, the stopping potential and the maximum electron speed.

Accepted Answer

Saturation current **doubles**, because twice as many photons arrive and each can free at most one electron.

Stopping potential is **unchanged**, since it depends only on $h
u-\phi$.

Maximum electron speed is likewise **unchanged**, for the same reason.

This separation of number from energy per photon is precisely what the classical wave picture cannot reproduce.

Question 4

A laser pointer of power $5$ mW emits at $650$ nm. Find the number of photons emitted per second and the force it exerts on a perfectly absorbing surface.

Accepted Answer

Photon energy $=\dfrac{1240}{650}=1.91$ eV $=3.05	imes10^{-19}$ J

$N=\dfrac{5	imes10^{-3}}{3.05	imes10^{-19}}=1.64	imes10^{16}$ per second

$p_{photon}=\dfrac{6.63	imes10^{-34}}{6.5	imes10^{-7}}=1.02	imes10^{-27}$ kg m s$^{-1}$

$F=Np=1.67	imes10^{-11}$ N, which equals $P/c$ as it must.

Question 5

A photon and an electron both have wavelength $2$ Å. Compare their momenta and their energies.

Accepted Answer

Momenta are **equal**: $p=\dfrac{6.63	imes10^{-34}}{2	imes10^{-10}}=3.32	imes10^{-24}$ kg m s$^{-1}$.

Photon: $E=\dfrac{12400}{2}=6200$ eV $=6.2$ keV.

Electron: $E=\dfrac{p^{2}}{2m}=\dfrac{\left(3.32	imes10^{-24}ight)^{2}}{2\left(9.11	imes10^{-31}ight)}=6.05	imes10^{-18}$ J $=37.8$ eV.

The photon carries about $160$ times more energy at the same wavelength.

Question 6

An alpha particle and a proton are accelerated through the same potential difference. Find the ratio of their de Broglie wavelengths.

Accepted Answer

$\lambda\propto\dfrac{1}{\sqrt{mq}}$ at fixed $V$.

For an alpha particle, $m=4m_p$ and $q=2e$, so $mq=8m_pe$.

$\dfrac{\lambda_\alpha}{\lambda_p}=\sqrt{\dfrac{m_pe}{8m_pe}}=\dfrac{1}{2\sqrt2}=0.354$

The alpha particle's wavelength is about $35\%$ of the proton's.

Question 7

Find the accelerating voltage that gives an electron the same de Broglie wavelength as a $10$ keV photon.

Accepted Answer

$\lambda=\dfrac{12400}{10000}=1.24$ Å

$\sqrt V=\dfrac{12.27}{1.24}=9.90$

$V=98$ V

The electron needs only about a hundred volts to match a ten kilovolt photon's wavelength — the practical reason electron microscopes are built rather than X-ray microscopes.

Question 8

Thermal neutrons at $300$ K are used for crystal diffraction. Find their de Broglie wavelength, taking $m=1.67	imes10^{-27}$ kg, and comment on its suitability.

Accepted Answer

$p=\sqrt{3\left(1.67	imes10^{-27}ight)\left(1.38	imes10^{-23}ight)(300)}=4.55	imes10^{-24}$ kg m s$^{-1}$

$\lambda=\dfrac{6.63	imes10^{-34}}{4.55	imes10^{-24}}=1.46$ Å

Typical atomic spacings are one to three angstroms, so the match is excellent. Neutrons at ordinary temperature diffract from crystals without any acceleration at all.

Question 9

In the Davisson-Germer experiment, $54$ eV electrons showed a diffraction peak at $50^{\circ}$ from a nickel crystal of spacing $0.91$ Å. Verify the de Broglie relation.

Accepted Answer

From de Broglie: $\lambda=\dfrac{12.27}{\sqrt{54}}=\dfrac{12.27}{7.35}=1.67$ Å

From Bragg, the $50^{\circ}$ scattering peak corresponds to a glancing angle $	heta=65^{\circ}$ on planes of spacing $d=0.91$ Å:

$\lambda=2d\sin	heta=2\left(0.91ight)\left(0.906ight)=1.65$ Å

The two independent routes agree to about one per cent, which is what established that electrons genuinely diffract.

Question 10

An X-ray tube operates at $40$ kV. Find the cut-off wavelength and the maximum photon energy, and state what happens to each if the current through the tube is doubled.

Accepted Answer

$\lambda_{min}=\dfrac{12400}{40000}=0.31$ Å

$E_{max}=40$ keV

Doubling the **current** doubles the intensity at every wavelength but leaves $\lambda_{min}$ and $E_{max}$ **completely unchanged**, since both depend only on the accelerating voltage.

This is the X-ray analogue of intensity not affecting the photoelectric stopping potential.

Question 11

The cut-off wavelength of an X-ray tube shifts from $0.5$ Å to $0.4$ Å. Find the change in operating voltage and the percentage change in maximum photon energy.

Accepted Answer

$V_1=\dfrac{12400}{0.5}=24800$ V; $V_2=\dfrac{12400}{0.4}=31000$ V

Increase $=6200$ V

$E_{max}$ rises from $24.8$ keV to $31.0$ keV, a rise of $25\%$ — the same percentage as the voltage, since the two are directly proportional.

Question 12

A double-slit experiment is run with a source so faint that photons arrive one at a time. Describe what is seen after a few seconds and after several hours, and state what this shows.

Accepted Answer

After a few seconds: a scattering of individual point-like flashes with no visible pattern, apparently random.

After several hours: the flashes have accumulated into the full interference pattern, fringe for fringe.

Since only one photon is present at a time, each must interfere with **itself**. The wave does not describe where the photon is but how likely it is to arrive at each point, and the same result has since been obtained with electrons, neutrons and whole molecules.

Practice — "Dual Nature of Matter and Radiation"

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