E.2 Quantum physics (HL)

Monochromatic light falls on the cathode of the photocell shown. The pd of the supply is adjusted and the current is measured. The intensity of the light is then doubled, with the frequency unchanged.

Which row gives the effect on the maximum (saturation) current and on the stopping potential?

Photons are scattered by free electrons. Which graph shows how the shift Δλ in the wavelength of the scattered photons varies with the scattering angle θ?

In a Compton scattering experiment, X-rays of wavelength λ are scattered by free electrons. A detector at a scattering angle of 60° measures a wavelength shift Δλ. The source is replaced by one that emits X-rays of wavelength 2λ, and the detector stays at 60°. What is the new wavelength shift?

A photocell is illuminated by monochromatic light, and the stopping potential is measured as V. The intensity of the light is tripled without changing its frequency. What are the new stopping potential and the new saturation photocurrent?

An automatic door sensor contains a photocell. Monochromatic light with a frequency above the threshold frequency falls on the photocathode. The intensity of the light is doubled and the frequency is kept constant. What happens to the maximum kinetic energy of the photoelectrons and to the saturation photoelectric current?

Which observation gives direct evidence that electrons have wave properties?

Ultraviolet light, with frequency above the threshold frequency of the metal, falls on a clean metal plate in the calibration unit of a space telescope. The intensity of the light is doubled and its frequency is unchanged. What happens to the number of photoelectrons emitted per second and to their maximum kinetic energy?

The cathode of a UV sensor at an athletics stadium is made of a metal with work function 2.5 eV. Use hc = 1240 eV nm. What is the longest wavelength of light that can cause photoelectric emission from this metal?

A beam of electrons passes through a thin graphite film and produces a pattern of concentric rings on a screen. What does this observation show?

Light of frequency f ejects photoelectrons of maximum kinetic energy E_k from a metal of work function Φ. The frequency of the light is doubled.

What is the new maximum kinetic energy?

An electron and a proton have the same kinetic energy.

What is the ratio (de Broglie wavelength of the electron) / (de Broglie wavelength of the proton)?

An X-ray photon of wavelength 10.0 pm is scattered by a free electron. The Compton wavelength of the electron is 2.43 pm.

What is the largest possible wavelength of the scattered photon?

Light of different frequencies falls on a metal surface, and the stopping potential is measured for each frequency. The graph shows the results.

What is the work function of the metal?

In the classical wave model of light, the energy of the light is spread evenly over each wavefront and an electron in a metal surface absorbs energy continuously from the wave.

Two experiments are carried out on the same clean metal surface:
Experiment 1: very dim light of a frequency well above the threshold frequency
Experiment 2: very intense light of a frequency below the threshold frequency

Which row gives the predictions of the classical wave model?

Ultraviolet radiation of wavelength 250 nm falls on a metal surface and the maximum kinetic energy of the emitted electrons is 1.20 eV. The radiation is replaced by radiation of wavelength 350 nm. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C)

What is the new maximum kinetic energy of the emitted electrons?

An electron and a photon both have a wavelength of 0.20 nm.

Which row compares the momentum and the energy of the photon with the momentum and the kinetic energy of the electron?

Monochromatic light of wavelength λ and power P falls on a photocathode. On average, only a fraction η of the incident photons each release one photoelectron, and every photoelectron released reaches the collector.

What is the photoelectric current?

Monochromatic light falls on a metal surface. The wavelength λ of the light is varied while the intensity is kept constant.

Which graph shows how the maximum kinetic energy E_max of the emitted electrons varies with λ?

A graph of stopping potential V_s against light frequency f is a straight line for photocathode metal X. Metal X is replaced by metal Y, which has a larger work function. How do the gradient and the intercept on the frequency axis change?

A flat solar sail faces the Sun and is perpendicular to the sunlight. At first its surface absorbs all incident light. The spacecraft then deploys a coating that reflects all the light straight back. At the same time, it moves to a position where the intensity of sunlight is half its original value. What is the ratio (new radiation force) / (original radiation force)?

In an electron diffraction tube, electrons pass through a thin graphite film and form rings on a screen. When the accelerating potential difference is 2.0 kV, a particular ring has radius r. The potential difference is increased to 8.0 kV. Assuming small diffraction angles and non-relativistic electrons, what is the new radius of this ring?

High-energy photons are Compton-scattered by free electrons in a radiation detector. At which scattering angle of the photon is the kinetic energy transferred to the recoiling electron greatest?

A small, isolated, initially uncharged metal sphere on a space probe is in a vacuum. It is illuminated with ultraviolet radiation. The photoelectrons are emitted with a maximum kinetic energy of 2.0 eV. The illumination continues for a long time. Which statement describes the final state of the sphere?

In a research laboratory, a beam of electrons and a beam of protons each have a kinetic energy of 5 keV. Relativistic effects can be ignored. The proton mass is about 1840 times the electron mass. What is the ratio \(\frac{\text{de Broglie wavelength of an electron}}{\text{de Broglie wavelength of a proton}}\)?

An electron source is made so weak that only one electron at a time is in the apparatus. Each electron passes a double slit and is recorded by a position-sensitive detector. Which statement describes what is observed?

A camera designer compares two photocathode materials, P and Q. The work function of Q is greater than that of P. For each material, the maximum kinetic energy of the photoelectrons is plotted against the frequency of the incident light. Which statement about the two straight lines is correct?

In a sprint-timing gate a photocell is illuminated by light of frequency f, and the maximum kinetic energy of the photoelectrons is 1.0 eV. When the light is replaced by light of frequency 2f, the maximum kinetic energy is 3.4 eV. What is the work function of the metal?

A proton and an electron are each accelerated from rest through the same potential difference. The mass of the proton is 1840 times the mass of the electron. What is the approximate value of (de Broglie wavelength of the proton) ÷ (de Broglie wavelength of the electron)?

Electrons accelerated through a potential difference produce interference fringes of separation 0.80 mm at a detector behind a double slit. The accelerating potential difference is increased by a factor of 4; the slit separation and the slit-to-detector distance are unchanged. What is the new fringe separation?

X-ray photons of wavelength 70 pm and gamma-ray photons of wavelength 2.0 pm are each scattered through 90° by free electrons that are initially at rest.

Which row compares the shift in wavelength of the scattered photons and the fraction of the incident photon energy that is transferred to the electron?

Electrons, accelerated from rest through a potential difference V, pass through a narrow slit. The first minimum of the diffraction pattern is at a small angle θ to the central direction.

The potential difference is reduced to V/4. What is the new angle of the first minimum?

The same photocathode is illuminated in turn by two monochromatic light sources, P and Q. The graph shows how the photoelectric current I varies with the potential V of the anode relative to the cathode. For both sources, each incident photon has the same probability of releasing a photoelectron.

Which row compares the frequency and the intensity of light Q with those of light P?

A thermal neutron and an electron have the same kinetic energy, and both are non-relativistic. The neutron mass is about 1840 times the electron mass. What is the approximate value of (de Broglie wavelength of electron) / (de Broglie wavelength of neutron)?

A photon and a non-relativistic electron have the same wavelength. Which statement is correct?

Electrons accelerated from rest through a potential difference V pass through a double slit. Fringes of spacing s form on a distant screen. The accelerating potential difference is increased to 4V, and the slit separation is halved. The slit-to-screen distance is unchanged and relativistic effects can be ignored. What is the new fringe spacing?

Light falls on a metal of work function 2.0 eV. The stopping potential is 1.0 V. The wavelength of the light is then halved. What is the new stopping potential?

A small, isolated metal sphere in space is initially uncharged and far from all other objects. Its work function is 4.3 eV. It is illuminated by ultraviolet light from a star, each photon having energy 5.6 eV. Photoelectrons are emitted until the emission stops. What is the final potential of the sphere relative to a point far away?

A photon of X-radiation scatters from a free electron at rest and is deflected through 60°. Use h = 6.63×10⁻³⁴ J s, m_e = 9.11×10⁻³¹ kg and c = 3.00×10⁸ m s⁻¹, with Δλ = (h/m_ec)(1 − cos θ). What is the increase in wavelength of the photon?

Monochromatic X-rays of wavelength 70.9 pm are directed at a block of graphite. A crystal spectrometer on a rotating arm measures the wavelength λ_f of the X-rays scattered through angle θ; each value is uncertain by ±0.15 pm. The Compton formula predicts that λ_f = λ_i + (h/m_ec)(1 − cos θ), where λ_i is the incident wavelength.

The graph shows λ_f against (1 − cos θ) with error bars and the line of best fit. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹)

In a vacuum photocell, light from a white source passes through one of six narrow-band filters and falls on a cathode coated with potassium. For each filter the student increases the reverse potential difference between anode and cathode until a sensitive ammeter in the circuit just reads zero; this is the stopping potential V_s. The frequency f passed by each filter is given by the manufacturer. Because the current falls to zero gradually, each V_s is uncertain by ±0.10 V.

The graph shows the data with error bars and the line of best fit. (e = 1.60 × 10⁻¹⁹ C)

In an electron diffraction tube, electrons are accelerated from rest through a potential difference V and then pass through a very thin layer of polycrystalline graphite. Rings appear on a screen a distance L = 0.135 m beyond it. For each V the student measures the radius r of the innermost ring. For small angles r = Lλ/d, where λ is the de Broglie wavelength of the electrons and d is the spacing of the planes of atoms that produce the ring.

The graph shows r against V^−1/2 with a line of best fit. (h = 6.63 × 10⁻³⁴ J s, m_e = 9.11 × 10⁻³¹ kg, e = 1.60 × 10⁻¹⁹ C)

A beam of fullerene molecules, known to be either C₆₀ (mass 720 u) or C₇₀ (mass 840 u), passes through a velocity selector and then through a single slit of width b = 0.20 μm. A detector scanned across the beam L = 1.50 m beyond the slit records a diffraction pattern; the distance y from its centre to the first minimum is measured for different molecular speeds v. The first minimum is at angle θ ≈ λ/b, so that y = (hL/mb)(1/v) for molecules of mass m.

The graph shows y against 1/v with the line of best fit. (h = 6.63 × 10⁻³⁴ J s, 1 u = 1.66 × 10⁻²⁷ kg)

A UV-index sensor for testing sunscreen uses a zinc photocathode, which has a work function of 4.30 eV. When ultraviolet radiation of a single wavelength falls on the cathode, the stopping potential is 1.20 V.
(h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, m_e = 9.11×10⁻³¹ kg, e = 1.60×10⁻¹⁹ C)

The photocathode of a night-vision image tube has a work function of 1.50 eV. Red light of wavelength 650 nm falls on the photocathode.
h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C, m_e = 9.11×10⁻³¹ kg.

A detector in a sports-science laboratory uses a metal cathode of work function 2.10 eV. Light of wavelength 450 nm is incident on the cathode. Use h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C and 1 eV = 1.60×10⁻¹⁹ J. Use unrounded values from earlier parts.

A rail engine and its wagons have a total mass of 1.8×10⁵ kg and travel at 20 m s⁻¹. Use h = 6.63×10⁻³⁴ J s.

A zinc sphere of radius 2.0 cm is mounted on an insulating stand in a vacuum. The photoelectric properties of zinc were first measured in a photocell: the graph shows the stopping potential V_s against the frequency f of the radiation incident on a zinc cathode.

(c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C, k = 8.99 × 10⁹ N m² C⁻²)

Neutrons from a research reactor pass through a block of heavy water at 320 K and emerge in thermal equilibrium with it. They are then diffracted by a crystal to reveal the arrangement of its atoms.

(k_B = 1.38 × 10⁻²³ J K⁻¹, mass of neutron = 1.675 × 10⁻²⁷ kg, m_e = 9.11 × 10⁻³¹ kg, h = 6.63 × 10⁻³⁴ J s, e = 1.60 × 10⁻¹⁹ C)

An ultraviolet flame detector contains a photocell whose metal cathode has a work function of 4.70 eV. The detector is used in a factory to confirm that a gas burner is alight. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C) (m_e = 9.11 × 10⁻³¹ kg)

An engineer tests a new photocathode for a night-vision image tube. The cathode is illuminated with monochromatic light of several frequencies, and the stopping potential V_s is measured each time.

f / 10¹⁴ Hz: 6.00, 7.00, 8.00, 9.00
V_s / V: 0.19, 0.60, 1.02, 1.43

(e = 1.60×10⁻¹⁹ C)

A materials scientist examines a thin polycrystalline film of a new battery-cathode material. Electrons are accelerated from rest through a potential difference of 4.00 kV and pass through the film. Rings appear on a fluorescent screen. The first ring is at an angle θ = 8.9° to the undeviated beam. Assume that this ring satisfies d sin θ = λ, where d is the spacing of the atomic planes.
(h = 6.63×10⁻³⁴ J s, m_e = 9.11×10⁻³¹ kg, e = 1.60×10⁻¹⁹ C)

A small demonstration spacecraft of total mass 5.00 kg uses a perfectly reflecting solar sail of area 80.0 m². Near Earth's orbit, the sail is held perpendicular to sunlight of intensity 1360 W m⁻². Ignore gravitational forces.
(h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹)

A semiconductor-wafer inspection facility can probe crystal structure with either electrons or neutrons. Both beams are prepared with a de Broglie wavelength of 0.150 nm.
(h = 6.63×10⁻³⁴ J s, m_e = 9.11×10⁻³¹ kg, m_n = 1.67×10⁻²⁷ kg, e = 1.60×10⁻¹⁹ C)

A laser power meter uses a vacuum photocell. The photocathode has a work function of 2.1 eV. A violet laser of wavelength 405 nm delivers 2.0 mW to the photocathode. Only 12% of the incident photons release a photoelectron, and all released electrons are collected.
h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C.

A transmission electron microscope in a hospital pathology laboratory is used to image viruses about 100 nm across. The electrons are accelerated from rest through a potential difference of 25 kV. Ignore relativistic effects.
h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C, m_e = 9.11×10⁻³¹ kg.

A light-gate timing system for sprint races uses a photocell. To characterise its photocathode, the stopping potential V_s is measured for light of different frequencies f:
f / 10¹⁴ Hz: 6.0, 7.0, 8.0, 9.0, 10.0
V_s / V: 0.18, 0.60, 1.01, 1.43, 1.84
e = 1.60×10⁻¹⁹ C.

A smartphone ambient-light sensor relies on electrons absorbing energy from incident light. In a simple model, the sensing surface has a work function of 2.3 eV. It is illuminated by dim light of intensity 1.0 W m⁻².
e = 1.60×10⁻¹⁹ C.

Electrons are accelerated from rest through a potential difference of 2.50 kV and then pass through a double slit with slit separation 5.0×10⁻⁷ m. A detector is placed 1.2 m beyond the slits. Use h = 6.63×10⁻³⁴ J s, m_e = 9.11×10⁻³¹ kg and e = 1.60×10⁻¹⁹ C. Ignore relativistic effects. The fringe separation is w = λD/s.

An X-ray photon of wavelength 2.50×10⁻¹¹ m from the gas around a binary star scatters from a free electron that is initially at rest. The photon is scattered through 120°. Use h = 6.63×10⁻³⁴ J s, m_e = 9.11×10⁻³¹ kg and c = 3.00×10⁸ m s⁻¹. The increase in photon wavelength is Δλ = (h/m_ec)(1 − cos θ).

A student tests a photocell for a telescope camera. The stopping potential V_s is measured for light of several frequencies f.
Frequency f / 10¹⁴ Hz: 6.0, 7.0, 8.0, 9.0, 10.0
Stopping potential V_s / V: 0.19, 0.60, 1.02, 1.43, 1.85
Use e = 1.60×10⁻¹⁹ C. The graph of V_s against f is a straight line.

A beam of neutrons, each with kinetic energy 0.025 eV, is incident on a crystal surface on which rows of atoms are 0.45 nm apart. Use h = 6.63×10⁻³⁴ J s, m_n = 1.675×10⁻²⁷ kg, m_e = 9.11×10⁻³¹ kg and 1 eV = 1.60×10⁻¹⁹ J. Use unrounded values from earlier parts.

A photodetector on a space telescope receives light of wavelength 500 nm at a total power of 3.0×10⁻¹² W. Of the photons arriving, 2.0% each release one photoelectron. Use h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C. Use unrounded values from earlier parts.

In a hospital gamma camera, a patient is given technetium-99m, which emits gamma photons of energy 140 keV. Some photons are Compton scattered by electrons in the patient's body before reaching the camera. Treat these electrons as free and at rest. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C) (m_e = 9.11 × 10⁻³¹ kg)

Electrons, accelerated from rest through a potential difference of 600 V, pass through two very narrow parallel slits whose centres are 2.5 × 10⁻⁷ m apart, cut in a thin membrane. A position-sensitive detector, 0.60 m beyond the slits, records the point of arrival of each electron as a single dot. The beam current is so small that the dots appear one at a time; after a long time they build up the distribution shown in the graph.

(h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C, m_e = 9.11 × 10⁻³¹ kg)

A solar sail is a large, very thin sheet that is accelerated by the momentum of sunlight. A sail of area 1600 m² has a total mass (sail and payload) of 12 kg. Sunlight strikes the sail at normal incidence and is completely reflected.

(L_☉ = 3.85 × 10²⁶ W, M_☉ = 1.99 × 10³⁰ kg, 1 AU = 1.50 × 10¹¹ m, G = 6.67 × 10⁻¹¹ N m² kg⁻², h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹)

A gamma camera in a hospital detects 140 keV photons emitted by a radioactive tracer inside a patient. Some photons are Compton-scattered by electrons in the patient's tissue before reaching the camera. Treat the electrons as free and at rest.
(h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, m_e = 9.11×10⁻³¹ kg, e = 1.60×10⁻¹⁹ C)

Neutrons from a research reactor are slowed in a moderator at 300 K. They are then used to study how hydrogen is distributed inside a fuel-cell membrane. Assume the neutrons' mean kinetic energy equals that of an ideal gas at the moderator temperature.
(k_B = 1.38×10⁻²³ J K⁻¹, m_n = 1.67×10⁻²⁷ kg, h = 6.63×10⁻³⁴ J s)

An airport security scanner uses Compton back-scattering. X-ray photons of energy 60.0 keV strike a suitcase, and the scanner detects photons scattered back through 180°. Treat the scattering electrons as free and at rest.
(h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, m_e = 9.11×10⁻³¹ kg, e = 1.60×10⁻¹⁹ C)

An isolated aluminium component on a space probe can be modelled as a conducting sphere of radius 0.50 m. It is initially uncharged. It is illuminated by solar ultraviolet radiation of wavelength 122 nm. The work function of aluminium is 4.1 eV. Ignore any charge collected from the surrounding space plasma.
h = 6.63×10⁻³⁴ J s, c = 3.00×10⁸ m s⁻¹, e = 1.60×10⁻¹⁹ C, ε₀ = 8.85×10⁻¹² F m⁻¹.