explain why the maximum kinetic energy of photoelectrons is independent of intensity, whereas the photoelectric current is proportional to intensity

Energy and Momentum of a Photon – Cambridge 9702 (Section 22.1)

Fundamental Relations

• Photon energy: \(E = h\nu = \dfrac{hc}{\lambda}\)
• Photon momentum: \(p = \dfrac{E}{c} = \dfrac{h}{\lambda}\)
• Photon mass: \(m_{\text{photon}} = 0\) (mass‑less particle)
• Planck’s constant: \(h = 6.626\times10^{-34}\ \text{J·s}\)
• Speed of light: \(c = 3.00\times10^{8}\ \text{m·s}^{-1}\)
• Useful conversion: \(1\ \text{eV}=1.602\times10^{-19}\ \text{J}\)

Wave–Particle Duality – Cambridge 9702 (Section 22.2 & 22.3)

Evidence for Wave Behaviour

• Interference (Young’s double‑slit)
• Diffraction (single‑slit, crystal lattices)
• Polarisation

Evidence for Particle Behaviour

• Photoelectric effect (Section 22.2)
• Compton scattering – photon‑electron collision where the photon’s wavelength increases:

  \[

  \lambda' - \lambda = \frac{h}{m_e c}(1-\cos\theta)

  \]

  demonstrating that photons carry momentum \(p=h/\lambda\).

• Photon‑counting experiments (e.g. photomultiplier tubes)

Photoelectric Effect – Cambridge 9702 (Section 22.2)

Key Observations Required by the Syllabus

1. The maximum kinetic energy of emitted electrons depends on the frequency (or wavelength) of the incident light, not on its intensity.
2. The photo‑electric current (number of electrons per second) is directly proportional to the light intensity.
3. No electrons are emitted if the photon frequency is below a material‑specific threshold frequency \(\nu0\) (equivalently, if the wavelength is longer than the threshold wavelength \(\lambda0\)).

Einstein’s Photoelectric Equation

\[

K_{\max}=h\nu-\phi

\]

• \(K_{\max}\) – maximum kinetic energy of the emitted electrons.
• \(\phi\) – work function of the metal (minimum energy needed to free an electron).
• Because \(\phi = h\nu0 = \dfrac{hc}{\lambda0}\), the equation can also be written as

  \[

  K{\max}=h(\nu-\nu0)=\frac{hc}{\lambda}-\frac{hc}{\lambda_0}.

  \]

Stopping (Retarding) Potential

To measure \(K{\max}\) a retarding potential \(Vs\) is applied so that the most energetic electrons are just prevented from reaching the collector:

\[

eVs = K{\max}\qquad\Longrightarrow\qquad eV_s = h\nu-\phi

\]

Plotting \(Vs\) (or \(eVs\)) against \(\nu\) gives a straight line; the slope equals \(h\) and the intercept equals \(-\phi/e\).

Why \(K_{\max}\) Is Independent of Light Intensity

• Intensity \(I\) is the energy delivered per unit area per unit time:

  \[

  I = \frac{P}{A}= \frac{N\,h\nu}{A\,t},

  \]

  where \(N\) is the number of photons arriving in time \(t\).

• Changing \(I\) changes the photon flux \(N/t\) but does not change the energy of each individual photon (\(h\nu\)).
• Each emitted electron can absorb at most the energy of one photon; the excess energy \(h\nu-\phi\) – and therefore \(K_{\max}\) – is fixed by the photon frequency alone.

Why the Photo‑electric Current Is Proportional to Intensity

The photo‑electric current is the charge flow caused by emitted electrons:

\[

I{\text{photo}} = e\,\frac{dNe}{dt}

\]

• Each photon can liberate at most one electron, so the emission rate \(\frac{dN_e}{dt}\) is proportional to the photon arrival rate \(\frac{N}{t}\).
• The photon arrival rate is directly proportional to the light intensity \(I\). Consequently,

  \[

  I_{\text{photo}}\propto I.

  \]

Threshold Wavelength \(\lambda_0\)

It is often convenient to express the threshold condition in terms of wavelength:

\[

\lambda_0 = \frac{hc}{\phi}.

\]

If the incident light has \(\lambda > \lambda0\) (i.e. \(\nu < \nu0\)), no electrons are emitted.

Typical Experimental Arrangement (AO2/AO3)

• Components: monochromatic light source (often a filtered lamp or laser), collimating lens, clean metal cathode, collector plate, variable retarding potential, sensitive ammeter.
• Procedure (summary):

  1. Set the light to a known frequency (or wavelength) and adjust the intensity.
  2. Vary the retarding potential until the measured current falls to zero – record this stopping potential \(V_s\).
  3. Repeat for at least three different frequencies.

• Typical sources of error:

  • Surface contamination or oxidation changing the work function \(\phi\).
  • Contact potentials between electrodes that add to the measured \(V_s\).
  • Inaccurate wavelength calibration of the light source.
  • Statistical fluctuations in photon flux (random error).

Data Analysis – Linear Regression (AO3 Example)

From the measured pairs \((\nui, V{s,i})\) a straight‑line fit to

\[

eV_s = h\nu - \phi

\]

is performed. The slope \(m\) gives an experimental value of \(h\) and the intercept \(c\) gives \(-\phi\).

Example calculation of uncertainties (using the least‑squares formulas):

\[

m = \frac{N\sum \nui V{s,i} - \sum \nui \sum V{s,i}}{N\sum \nui^2 - (\sum \nui)^2},

\qquad

\sigmam = \sqrt{\frac{1}{N-2}\,\frac{\sum (V{s,i} - m\nui - c)^2}{\sum \nui^2 - (\sum \nu_i)^2/N}}.

\]

Multiplying the slope by the elementary charge \(e\) yields \(h\) (in J·s) and the standard error \(\sigma_m e\) gives the uncertainty in \(h\).

de Broglie Wavelength – Wave Nature of Matter (Section 22.3)

de Broglie Relation

\[

\lambda = \frac{h}{p}

\]

• For a non‑relativistic electron accelerated through a potential difference \(V\):

  \[

  p = \sqrt{2m_e eV}\quad\Longrightarrow\quad

  \lambda = \frac{h}{\sqrt{2m_e eV}}.

  \]

• Electron diffraction experiments (Davisson–Germer, electron diffraction gratings) confirm the predicted wavelengths, providing further evidence of wave‑particle duality.

Atomic Energy Levels & Line Spectra – Cambridge 9702 (Section 22.4)

Quantised Energy Levels

When an electron in an atom moves from a higher level \(Ei\) to a lower level \(Ef\) a photon is emitted:

\[

h\nu = Ei - Ef.

\]

Hydrogen Series (examples of syllabus‑required series)

where \(R_H = 1.097\times10^{7}\ \text{m}^{-1}\) is the Rydberg constant.

Emission vs. Absorption Spectra

• Emission spectrum: bright lines on a dark background – produced when excited atoms return to lower levels.
• Absorption spectrum: dark lines on a continuous background – produced when a continuous source passes through a cool gas that absorbs specific wavelengths.

Applications (selected)

• Photo‑electric cells – light meters, solar panels (photovoltaic effect).
• Compton scattering – used in medical imaging (CT) and astrophysics.
• X‑ray production – high‑energy photons generated by accelerating electrons onto a metal target.
• Spectroscopy – element identification via characteristic line spectra.
• Electron microscopy – exploits the short de Broglie wavelength of high‑energy electrons for sub‑nanometre resolution.

Summary Table – Intensity vs. Kinetic Energy

Key Take‑aways

• Photons are mass‑less particles with quantised energy \(E=h\nu\) and momentum \(p=h/\lambda\).
• The photoelectric effect shows that the maximum kinetic energy of emitted electrons is set by the excess photon energy over the work function:

  \[

  K_{\max}=h\nu-\phi,

  \]

  and is independent of intensity.

• The photo‑electric current reflects how many photons strike the surface per second; therefore it is directly proportional to the light intensity.
• Compton scattering and the photoelectric effect provide the particle‑evidence required by the syllabus, while interference, diffraction and electron diffraction provide the wave‑evidence.
• de Broglie’s relation \(\lambda=h/p\) extends wave‑particle duality to matter, and electron diffraction experiments confirm it.
• Atomic line spectra (Lyman, Balmer, Paschen, Brackett…) demonstrate quantised energy levels; the relation \(h\nu=Ei-Ef\) links spectroscopy to photon energy.
• Understanding these concepts is essential for AO1 (knowledge), AO2 (application) and AO3 (analysis/evaluation) in the Cambridge 9702 syllabus.