Dawn of modern physics: Class 12 Physics Notes, Short Questions and Numericals

Black body radiation and Planck's quantum idea, the photoelectric effect and Einstein's equation, the Compton effect, pair production, de Broglie waves, the electron microscope, the uncertainty principle, and relativity.

Key points

At the end of the 19th century classical physics could not explain black body radiation, the photoelectric effect and atomic spectra. These problems led to quantum theory and relativity. A black body is a perfect absorber and a perfect emitter of radiation. The spectrum of its radiation depends only on temperature. As the temperature rises, the peak moves to shorter wavelength (Wien's displacement law, λmax T = constant) and the total power rises as T⁴ (Stefan's law).

Classical theory predicted that intensity would become infinite at short wavelengths, which does not happen. Planck solved this in 1900 by assuming that energy is emitted and absorbed in small packets called quanta. The energy of one quantum is E = hf, where h = 6.63 × 10⁻³⁴ J s is Planck's constant and f is the frequency.

Photoelectric effect: when light of high enough frequency falls on a metal surface, electrons are emitted. These are called photoelectrons. The experimental facts are:

Facts about photoelectric emission:

Below a certain threshold frequency f₀ no electrons come out, however bright the light.
Above f₀ emission is instant, with no delay.
The maximum kinetic energy of photoelectrons depends on the frequency, not on the intensity.
The number of photoelectrons per second, and so the photocurrent, is proportional to the intensity of light.

Wave theory could not explain these facts. Einstein explained them in 1905 using photons. Light travels as photons of energy hf. One photon gives all its energy to one electron. Part of this energy, the work function φ, is needed to free the electron from the metal, and the rest is its kinetic energy: hf = φ + KEmax. The work function is φ = hf₀ and the threshold wavelength is λ₀ = c/f₀. The energy of an electron is often given in electron volts (eV), where 1 eV = 1.6 × 10⁻¹⁹ J.

The stopping potential V₀ is the reverse voltage that just stops the fastest photoelectrons: eV₀ = KEmax. A graph of KEmax against frequency is a straight line with gradient h and intercept on the frequency axis at f₀. Photocells use this effect in burglar alarms, automatic doors and light meters.

Photons also have momentum, p = h/λ = E/c. The Compton effect supports this. When X-rays collide with free electrons, the scattered X-rays have a longer wavelength, because the photon gives some energy to the electron. It is explained by treating the collision as one between a photon and an electron, with conservation of energy and momentum. The shift is Δλ = (h/mc)(1 − cosθ), where h/mc = 2.43 × 10⁻¹² m.

Pair production: a photon of high energy passing near a nucleus can change into an electron and a positron (the antiparticle of the electron). The photon energy must be at least 2mc² = 1.02 MeV. The reverse process, annihilation, occurs when an electron and a positron meet and give two gamma photons.

De Broglie suggested in 1924 that matter also behaves like a wave. A particle of momentum p has the wavelength λ = h/p = h/(mv). The wavelength is only noticeable for tiny particles such as electrons. It was confirmed by electron diffraction experiments in crystals. This wave nature is used in the electron microscope. Because fast electrons have a wavelength far smaller than visible light, the microscope can resolve much finer details.

Heisenberg's uncertainty principle states that the position and momentum of a particle cannot both be known exactly at the same time: Δx Δp ≥ h/(4π). Likewise ΔE Δt ≥ h/(4π). This is not due to poor instruments but to the wave particle nature of matter.

Special relativity (Einstein, 1905) rests on two postulates:

The laws of physics are the same in all inertial frames.
The speed of light in vacuum is the same for all observers, c = 3.0 × 10⁸ m/s.
Results: time dilation (a moving clock runs slow), length contraction along the direction of motion, and increase of mass with speed. Mass and energy are linked by E = mc².

Study tip: Learn Einstein's equation and be ready to draw the KEmax against f graph, labelling the gradient and intercepts. Keep a list of constants (h, c, e, electron mass) at the top of your answer sheet. Check powers of ten twice in every numerical.

Definitions and flashcards

Say the meaning first, then open the card to check.

Black body

A body that absorbs all radiation falling on it and emits a spectrum that depends only on temperature.

Quantum

The smallest packet of energy, hf, that can be emitted or absorbed.

Photon

A quantum of electromagnetic radiation with energy hf.

Work function

The minimum energy needed to free an electron from a metal surface.

Threshold frequency

The lowest frequency of light that can cause photoelectric emission.

Stopping potential

The reverse voltage that stops the fastest photoelectrons.

Electron volt

The energy gained by an electron through 1 V, equal to 1.6 × 10⁻¹⁹ J.

De Broglie wavelength

The wavelength λ = h/p associated with a moving particle.

Time dilation

The slowing of a moving clock as seen by a stationary observer.

Formula sheet

Name	Formula	What it means
Photon energy	E = hf = hc / λ	h = 6.63 × 10⁻³⁴ J s.
Einstein's equation	hf = φ + KEmax	φ is the work function.
Work function	φ = hf₀	f₀ is the threshold frequency.
Stopping potential	eV₀ = KEmax	e = 1.6 × 10⁻¹⁹ C.
Photon momentum	p = h / λ	Also p = E / c.
De Broglie wavelength	λ = h / (mv)	For a particle of mass m and speed v.
Compton shift	Δλ = (h / mc)(1 − cosθ)	θ is the scattering angle.
Mass energy	E = mc²	c = 3.0 × 10⁸ m/s.
Uncertainty	Δx Δp ≥ h / 4π	Heisenberg's principle.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

State Planck's quantum hypothesis. (2 marks)

Energy is emitted or absorbed by a body only in whole number multiples of a small packet called a quantum. The energy of one quantum is E = hf, where h is Planck's constant and f the frequency of the radiation.

Define the work function and threshold frequency. (2 marks)

The work function is the minimum energy needed to remove an electron from the surface of a metal. The threshold frequency is the lowest frequency of light that can cause photoelectric emission from that metal, and φ = hf₀.

Why can the wave theory of light not explain the photoelectric effect? (3 marks)

Wave theory predicts that bright light of any frequency should eventually free electrons, that emission should be delayed while the energy builds up, and that the electron energy should rise with intensity. Experiments show a threshold frequency, instant emission, and electron energy that depends only on frequency. These facts are explained by photons.

What is pair production? (2 marks)

A gamma photon of energy at least 1.02 MeV, passing close to a nucleus, changes into an electron and a positron. Energy is converted into mass by E = mc².

Explain the de Broglie hypothesis and why we do not see the wave nature of a cricket ball. (3 marks)

De Broglie proposed that every moving particle has a wave of wavelength λ = h/(mv). For a cricket ball the mass is large, so mv is large and the wavelength is extremely small, far less than anything we can detect. For an electron the mass is tiny and the wavelength is of the order of atomic spacing, so diffraction can be seen.

Describe Einstein's explanation of the photoelectric effect and how it explains the threshold frequency. (5 marks)

Einstein proposed that light travels in packets called photons, each with energy hf. One photon is absorbed by one electron in the metal. Part of the photon energy, the work function φ, is used to free the electron and the remainder appears as kinetic energy, so hf = φ + KEmax. If hf is less than φ the electron cannot escape, whatever the number of photons. So there must be a threshold frequency f₀ = φ/h below which there is no emission. Above it the electron leaves at once, because one photon gives all its energy in a single event. The maximum kinetic energy rises with frequency. Higher intensity means more photons and so more electrons per second, not more energy per electron.

Numericals with worked steps

Find the energy in joules of a photon of frequency 6.0 × 10¹⁴ Hz (h = 6.63 × 10⁻³⁴ J s).

E = hf = 6.63 × 10⁻³⁴ × 6.0 × 10¹⁴.
6.63 × 6.0 = 39.78.
Powers: 10⁻³⁴ × 10¹⁴ = 10⁻²⁰, so E = 39.78 × 10⁻²⁰ = 3.978 × 10⁻¹⁹ J.

Answer: E ≈ 3.98 × 10⁻¹⁹ J (about 2.5 eV)

The work function of a metal is 2.0 eV. Light of energy 3.5 eV falls on it. Find the maximum kinetic energy and the stopping potential.

KEmax = hf − φ = 3.5 − 2.0 = 1.5 eV.
In joules: 1.5 × 1.6 × 10⁻¹⁹ = 2.4 × 10⁻¹⁹ J.
eV₀ = KEmax, so V₀ = 1.5 V.

Answer: KEmax = 1.5 eV = 2.4 × 10⁻¹⁹ J, V₀ = 1.5 V

Find the de Broglie wavelength of an electron moving at 2.0 × 10⁶ m/s (m = 9.1 × 10⁻³¹ kg).

λ = h/(mv).
mv = 9.1 × 10⁻³¹ × 2.0 × 10⁶ = 1.82 × 10⁻²⁴.
λ = 6.63 × 10⁻³⁴ / 1.82 × 10⁻²⁴ = 3.64 × 10⁻¹⁰ m.

Answer: λ ≈ 3.6 × 10⁻¹⁰ m

How much energy is released if 1.0 × 10⁻³ kg of matter is completely converted into energy?

E = mc² = 1.0 × 10⁻³ × (3.0 × 10⁸)².
(3.0 × 10⁸)² = 9.0 × 10¹⁶.
E = 9.0 × 10¹³ J.

Answer: E = 9.0 × 10¹³ J

Common mistakes

Saying that brighter light gives photoelectrons more energy. Only the number of electrons increases.
Mixing eV and joules. Convert with 1 eV = 1.6 × 10⁻¹⁹ J before using SI formulas.
Writing de Broglie wavelength as h × mv. It is h divided by mv.
Claiming the uncertainty principle comes from faulty instruments. It is a basic property of nature.
Forgetting that the threshold frequency belongs to the metal, not the light.

Practise this chapter

Practice questions and timed test Printable worksheet