1.
A point source of monochromatic light of $1.0 \text{ mW}$ is emitting photons of energy $5.0 \text{ eV}$ each. It is placed $0.5 \text{ m}$ away from a metal surface of work function $2.0 \text{ eV}$ and area $1.0 \text{ cm}^2$. If the efficiency of photoelectron emission is 1%, calculate the number of photoelectrons emitted per second.
2.
Light of frequency $1.5\nu_0$ is incident on a photosensitive material. If the frequency is halved and intensity is doubled, what happens to the photoelectric current? Provide a detailed mathematical reason.
3.
A beam of light consists of two wavelengths, $4000 \text{ \AA}$ and $4800 \text{ \AA}$, having a total intensity of $1.0 \times 10^{-2} \text{ W/m}^2$ equally distributed between them. The beam falls on an area of $1.0 \text{ cm}^2$ of a magnesium surface (work function $3.7 \text{ eV}$). Calculate the number of photoelectrons emitted per second.
4.
The stopping potential $V_0$ for a metal is measured at several frequencies. The data is plotted as $V_0$ versus $\nu$. If the intercept on the frequency axis is $5 \times 10^{14} \text{ Hz}$ and the intercept on the potential axis is $-2.07 \text{ V}$, verify the value of Planck's constant.
5.
AI Image Prompt: A high-quality, mathematically correct landscape graph showing the variation of Stopping Potential (V0) on the y-axis with the Reciprocal of Wavelength (1/lambda) on the x-axis for a metal. The graph must show a straight line with a negative intercept on the y-axis and a positive intercept on the x-axis. Label the slope as hc/e. The background of the whole image should be fully white.
Filename: Level3_Q5_Stopping_Potential_vs_Reciprocal_Wavelength.png
6.
An electron and a photon have the same energy $E$. Prove that the ratio of the de Broglie wavelength of the electron to the wavelength of the photon is $\sqrt{E/2m_ec^2}$.
7.
Calculate the de Broglie wavelength of an electron moving with 1% of the speed of light. Take into account the relativistic correction if necessary (discuss when it becomes mandatory).
8.
A proton and an $\alpha$-particle are accelerated through the same potential difference $V$. Find the ratio of their (i) momenta, and (ii) de Broglie wavelengths.
9.
Find the de Broglie wavelength of a thermal neutron at room temperature $27^\circ\text{C}$. Compare this with the wavelength of an electron accelerated through $150 \text{ V}$.
10.
AI Image Prompt: A clean, high-quality landscape diagram of the Davisson-Germer experiment setup specifically showing the Electron Gun, the Nickel crystal tilted at an angle, and the movable detector. Include a zoomed-in callout showing the atomic planes of Nickel and the diffraction of the electron wave. The background of the whole image should be fully white.
Filename: Level3_Q10_Davisson_Germer_Detailed_Mechanism.png
11.
In the Davisson-Germer experiment, the intensity maximum was observed at $\phi = 50^\circ$. If the interplanar spacing of Nickel is $d = 0.91 \text{ \AA}$, calculate the experimental wavelength and the percentage error relative to the theoretical de Broglie wavelength at $54 \text{ V}$.
12.
If the uncertainty in the position of an electron is equal to its de Broglie wavelength, prove that the uncertainty in its velocity is at least $1/4\pi$ times its velocity.
13.
A metallic surface is illuminated with radiation of wavelength $\lambda$. The stopping potential is $V$. If the same surface is illuminated with radiation of wavelength $2\lambda$, the stopping potential is $V/4$. Find the threshold wavelength for the surface.
14.
Determine the de Broglie wavelength of an electron orbiting in the ground state of a hydrogen atom ($r = 0.53 \text{ \AA}$). Does this support Bohr's postulate of quantized orbits?
15.
A $100 \text{ W}$ sodium lamp radiates energy uniformly in all directions. At what rate are photons passing through the pupil of an eye (area $0.4 \text{ cm}^2$) placed $2 \text{ m}$ away? ($\lambda_{Na} = 589 \text{ nm}$)
16.
Calculate the pressure exerted by a monochromatic light beam of intensity $I$ on a surface which has an absorption coefficient $a$ and reflection coefficient $r$.
17.
An electron is trapped in a one-dimensional box of length $L = 1 \text{ \AA}$. Calculate its ground state energy in eV using de Broglie’s relation.
18.
The work function of a surface is $2.5 \text{ eV}$. Find the maximum wavelength of light that can eject an electron with a de Broglie wavelength of $1 \text{ nm}$.
19.
AI Image Prompt: A clean landscape graph plotting the square of the maximum velocity of photoelectrons (v_max^2) on the y-axis versus the frequency (nu) of incident radiation on the x-axis. Show the threshold frequency nu0 and the linear relationship. Label the slope as 2h/m. The background of the whole image should be fully white.
Filename: Level3_Q19_VelocitySq_vs_Frequency.png
20.
How would the Davisson-Germer polar graph change if the target was a liquid surface instead of a single crystal? Explain the importance of periodic structures in matter-wave diffraction.
21.
A photon of wavelength $0.1 \text{ \AA}$ collides with an electron at rest and is scattered at $90^\circ$. Calculate the change in wavelength of the photon (Compton Effect - standard challenge).
22.
A non-relativistic particle of mass $m$ has a de Broglie wavelength $\lambda$. If its kinetic energy increases by a factor of 4, find the percentage change in its wavelength.
23.
In a photoelectric experiment, the stopping potential is $V_1$ for frequency $\nu_1$ and $V_2$ for frequency $\nu_2$. Derive the formula for Planck's constant $h$ and work function $\Phi$ in terms of these variables.
24.
A radio transmitter operates at a frequency of $880 \text{ kHz}$ and a power of $10 \text{ kW}$. Find the number of photons emitted per second. Why can we ignore the "particle" nature of radio waves?
25.
Calculate the threshold wavelength of a photon required to produce an electron-positron pair (rest mass of each is $0.511 \text{ MeV}$).
26.
Show that the group velocity of a matter wave is equal to the particle velocity, whereas the phase velocity can exceed the speed of light.
27.
Light of intensity $1.5 \times 10^{-2} \text{ W/m}^2$ falls on a sodium surface of area $2 \text{ cm}^2$. If the work function is $2.3 \text{ eV}$, how many photoelectrons are emitted per second? (Assume 100% quantum efficiency).
28.
If an electron is accelerated through $10 \text{ kV}$, calculate its velocity. Is it necessary to use relativistic mass $m = m_0 / \sqrt{1 - v^2/c^2}$ here? Calculate the error if $m_0$ is used.
29.
Derive the de Broglie wavelength of an electron in terms of its kinetic energy $K$, taking relativistic effects into account.
30.
A laser beam of power $P$ and wavelength $\lambda$ is incident on a solar cell. If the cell converts each photon into an electron-hole pair with efficiency $\eta$, find the short-circuit current produced.
31.
The work function of Potassium is $2.3 \text{ eV}$. What is the stopping potential when it is illuminated by light of $400 \text{ nm}$?
32.
Prove that the product of the de Broglie wavelength and the frequency of a photon is the speed of light, but for a material particle, it is not.
33.
A particle is moving in a circular path of radius $R$ with constant speed. Is its de Broglie wavelength constant? Justify.
34.
For a metal, the graph of $K_{max}$ vs $\nu$ has an intercept of $4 \times 10^{14} \text{ Hz}$. If light of $300 \text{ nm}$ is used, find the maximum speed of electrons.
35.
An $\alpha$-particle and a deuteron have the same de Broglie wavelength. Find the ratio of their kinetic energies.
36.
In the Davisson-Germer experiment, why does the electron detector need to be highly sensitive to current? Calculate the current if $10^{12}$ electrons are collected per second.
37.
What Accelerating Potential is needed to give an electron a wavelength of $0.5 \text{ \AA}$?
38.
Explain the term 'Work Function' from the perspective of potential wells in solid-state physics.
39.
A photon and an electron have the same wavelength of $2 \text{ nm}$. Find the ratio of their momenta.
40.
A $5 \text{ mW}$ laser beam of wavelength $633 \text{ nm}$ is pointed at a target. Find the number of photons hitting the target every minute.
41.
Show that for a photon, the energy $E = pc$, where $p$ is momentum. Use this to find the momentum of a $600 \text{ nm}$ photon.
42.
If the frequency of light incident on a metal is doubled, does the maximum kinetic energy of the photoelectrons double? Explain.
43.
Calculate the de Broglie wavelength of an electron in the first excited state ($n=2$) of a $He^+$ ion.
44.
The glancing angle in Davisson-Germer experiment is $65^\circ$. If $d = 0.91 \text{ \AA}$, what was the potential $V$ used?
45.
Determine the maximum kinetic energy of a photoelectron if the stopping potential is $1.5 \text{ V}$. Express in Joules.
46.
A beam of electrons with speed $v$ is incident on a slit of width $d$. Describe the diffraction pattern observed if $h/mv \approx d$.
47.
The work function of Aluminum is $4.2 \text{ eV}$. If light of $2000 \text{ \AA}$ is incident, what is the stopping potential?
48.
Find the de Broglie wavelength of a $1 \text{ mg}$ grain of sand moving at $1 \text{ mm/s}$. Compare with atomic dimensions.
49.
What happens to the threshold frequency of a metal if it is heated? (Consider the change in work function with temperature).
50.
A particle of mass $M$ at rest decays into two particles of masses $m_1$ and $m_2$ with non-zero velocities. Find the ratio of their de Broglie wavelengths.
51.
Calculate the ratio of the de Broglie wavelength of a $1 \text{ keV}$ electron to a $1 \text{ keV}$ photon.
52.
A photoelectric surface has a stopping potential of $3 \text{ V}$ for wavelength $\lambda$ and $1 \text{ V}$ for wavelength $2\lambda$. Find the threshold wavelength.
53.
In the Davisson-Germer experiment, if the peak shifts to $45^\circ$, how must the voltage have changed?
54.
Calculate the wavelength of an electron that has a momentum equal to that of a $500 \text{ nm}$ photon.
55.
Does a stationary charge have a de Broglie wavelength? Support your answer using the $\lambda = h/p$ relation.
56.
Explain how the "cut-off" wavelength of X-rays is related to the photoelectric effect in reverse.
57.
Find the kinetic energy of an electron whose de Broglie wavelength is $10^{-12} \text{ m}$.
58.
A $10 \text{ W}$ light source emitting at $500 \text{ nm}$ is $100\%$ efficient. Find the force it exerts on a black surface.
59.
Identify the particle whose de Broglie wavelength is $0.286 \text{ \AA}$ when accelerated through $100 \text{ V}$.
60.
State the physical consequence of the fact that $h$ is very small in the macroscopic world.
