Photons

SYLLABUS

Quantum nature of radiation, Einstein’s photoelectric equation; Stopping potential, Measurement of Planck’s constant

What is This Chapter About?

By the end of the 19th century, classical physics seemed complete. Newton’s mechanics explained motion. Maxwell’s equations explained electricity, magnetism, and light. Thermodynamics explained heat. Yet three phenomena stubbornly resisted explanation by these classical theories: blackbody radiation, the photoelectric effect, and atomic line spectra. The resolution of these puzzles required a completely new physics — quantum mechanics — and the key insight that launched it was the concept of the photon: the idea that light energy is not continuous but comes in discrete packets.

Chapter 21 of NEB Grade 12 Physics explores two of these quantum phenomena: the photoelectric effect and its explanation through Einstein’s photon theory. Carrying 3 teaching hours, this chapter marks the transition from classical to modern physics in the Grade 12 course and introduces the wave-particle duality of light — one of the most profound and counterintuitive ideas in the history of science.

Key Concepts You Will Learn

Quantum Nature of Radiation

In 1900, Max Planck made a revolutionary hypothesis to explain blackbody radiation: energy is not emitted or absorbed continuously but in discrete packets called quanta. Each quantum of electromagnetic radiation (light) carries energy:

[latex]E = hf = \frac{hc}{\lambda}[/latex]

where h = 6.626 × 10⁻³⁴ J s is Planck’s constant, f is the frequency of the radiation, c is the speed of light, and λ is the wavelength. This equation is one of the most fundamental in all of modern physics. It shows that high-frequency radiation (short wavelength — UV, X-rays, gamma rays) carries high-energy quanta, while low-frequency radiation (long wavelength — infrared, radio waves) carries low-energy quanta.

Einstein extended Planck’s idea in 1905 by proposing that light is not just emitted and absorbed in quanta — it actually consists of discrete particles of electromagnetic energy called photons. Each photon carries energy [latex]E = hf[/latex], travels at the speed of light, has zero rest mass, and has momentum [latex]p = \frac{h}{\lambda} = \frac{hf}{c}[/latex].

Properties of Photons

Photons have several remarkable properties that distinguish them from classical waves and from ordinary matter particles. They travel at exactly the speed of light c in vacuum — always, for all observers (a result of special relativity). They have zero rest mass but carry energy and momentum. They are not deflected by electric or magnetic fields (they carry no charge). The number of photons determines the intensity of light, while the frequency determines the energy per photon — these are completely independent properties. A dim blue light has fewer photons but more energy per photon than a bright red light.

Work Function and the Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency falls on it. This phenomenon was discovered experimentally but completely defied classical explanation — classical wave theory predicted that any frequency of light should eventually eject electrons if intense enough, and that higher intensity should give electrons more energy. Both predictions were wrong.

The experimental observations were:

Electrons are only emitted if the light frequency exceeds a minimum threshold frequency [latex]f_0[/latex] — regardless of intensity. Below [latex]f_0[/latex], no electrons are emitted no matter how bright the light.

The maximum kinetic energy of emitted electrons depends only on the frequency of light — not on intensity. Higher frequency means more energetic electrons.

The number of emitted electrons per second is proportional to the light intensity — more photons eject more electrons, but each electron’s energy is determined by individual photon energy.

Emission is essentially instantaneous — there is no time delay waiting for energy to build up, even for very dim light.

The work function ([latex]\phi[/latex] or [latex]W_0[/latex]) of a metal is the minimum energy required to free an electron from its surface. Each metal has a characteristic work function that determines its threshold frequency: [latex]hf_0 = \phi[/latex].

Einstein’s Photoelectric Equation

Einstein explained all experimental observations with a single elegant equation in 1905 — work that earned him the Nobel Prize in 1921. His insight was that light consists of photons, and each photon interacts with a single electron. The photon’s energy goes partly to overcoming the work function (releasing the electron from the metal) and the rest becomes kinetic energy of the emitted electron:

[latex]hf = \phi + \frac{1}{2}mv_{max}^2[/latex]

Or equivalently: [latex]KE_{max} = hf – \phi = hf – hf_0 = h(f – f_0)[/latex]

This equation explains every observation: below threshold frequency, [latex]hf < \phi[/latex] so no electrons are emitted regardless of intensity. Above threshold, kinetic energy increases linearly with frequency. Intensity determines the number of photons (and hence electrons emitted) but not their individual energy. Emission is immediate because each photon transfers its energy to one electron instantly.

Stopping Potential

The maximum kinetic energy of emitted photoelectrons can be measured using the stopping potential [latex]V_s[/latex] — the reverse voltage needed to stop all emitted electrons:

[latex]eV_s = \frac{1}{2}mv_{max}^2 = hf – \phi[/latex]

A graph of stopping potential versus frequency is a straight line with slope [latex]\frac{h}{e}[/latex] and x-intercept at the threshold frequency [latex]f_0[/latex]. This graph allows Planck’s constant h to be determined experimentally from photoelectric measurements.

Millikan’s Verification of Einstein’s Equation

Robert Millikan (who also measured the elementary charge) performed careful photoelectric experiments between 1906 and 1916, precisely measuring the stopping potential for different frequencies of light for several metals. Plotting [latex]V_s[/latex] versus f gave straight lines with the same slope for all metals — confirming Einstein’s equation and giving a measured value of Planck’s constant in excellent agreement with Planck’s theoretical value. Millikan’s experiment was decisive experimental proof of the photon theory of light and Einstein’s photoelectric equation.

Wave-Particle Duality of Light

The photoelectric effect demonstrates that light behaves as particles (photons) — discrete energy packets that interact one-at-a-time with individual electrons. Yet interference and diffraction (Chapters 11 and 12) demonstrate that light behaves as waves — producing interference patterns that particles cannot explain.

This apparent contradiction is resolved by wave-particle duality — light is neither purely a wave nor purely a particle, but has aspects of both. Which aspect manifests depends on the experimental situation: wave behaviour appears in propagation and interference; particle behaviour appears in interaction with matter (absorption and emission). This duality is not a deficiency of our understanding but a fundamental feature of quantum reality.

Why This Chapter Matters for NEB Exams

Photons is examined consistently in NEB Grade 12 Physics. Einstein’s photoelectric equation — stating it, explaining each term, and solving numerical problems — appears in virtually every board paper. Calculating threshold frequency, work function, maximum kinetic energy of emitted electrons, and stopping potential are standard numerical problems. The properties of photons — energy, momentum, speed — and the quantum nature of radiation are standard short-answer topics. Millikan’s verification of Einstein’s equation and the concept of wave-particle duality are popular conceptual questions. Problems involving the relationship [latex]E = hf = \frac{hc}{\lambda}[/latex] and converting between frequency, wavelength, and photon energy appear regularly.

Important Topics to Focus On

• Quantum nature of radiation — Planck’s hypothesis
• Photon energy: [latex]E = hf = \frac{hc}{\lambda}[/latex]
• Photon momentum: [latex]p = \frac{h}{\lambda}[/latex]
• Properties of photons — speed, mass, charge, energy
• Photoelectric effect — experimental observations and why classical theory fails
• Work function [latex]\phi[/latex] and threshold frequency [latex]f_0[/latex]: [latex]\phi = hf_0[/latex]
• Einstein’s photoelectric equation: [latex]hf = \phi + KE_{max}[/latex]
• Stopping potential: [latex]eV_s = KE_{max} = hf – \phi[/latex]
• Graph of stopping potential vs frequency — slope = h/e, x-intercept = f₀
• Millikan’s verification of Einstein’s equation and measurement of h
• Wave-particle duality of light — reconciling interference and photoelectric effect
• Numerical problems on photon energy, photoelectric equation, and stopping potential

Connection to Other Chapters

Photons builds on Grade 11 Chapter 12 (Rate of Heat Flow) — blackbody radiation introduced Stefan’s Law and ideal radiators, which historically motivated Planck’s quantum hypothesis. Grade 12 Chapters 11 and 12 (Interference and Diffraction) established the wave nature of light — this chapter now reveals its complementary particle nature. The electron volt unit from Grade 11 Chapter 21 (Potential) is used extensively here for work functions and photon energies. Chapter 22 (Semiconductor Devices) connects through the photoelectric effect in solar cells — photons absorbed in a semiconductor liberate electron-hole pairs, generating current. Chapter 23 (Quantization of Energy) extends quantum ideas to atomic structure — the Bohr model uses photon emission and absorption to explain atomic line spectra. The photon momentum [latex]p = \frac{h}{\lambda}[/latex] connects to de Broglie’s hypothesis in Chapter 23 — where matter particles are assigned wavelengths by the same formula.