Dawn of Modern Physics notes

MDCAT Physics

This chapter explains the quantum nature of radiation, photons, the photoelectric effect, X-rays, pair production and the wave particle duality of matter and light. It also introduces de Broglie waves, special relativity and the relation between mass and energy.

Need for Modern Physics and Quantum Theory

Classical physics explains the motion of ordinary-sized objects and many wave phenomena. It could not fully explain blackbody radiation, the photoelectric effect and the spectra of atoms. These observations led to modern physics.

According to Planck's quantum theory, energy is exchanged between matter and radiation in separate packets called quanta. The energy of one quantum is proportional to the frequency of radiation.

A quantum is a discrete packet of energy exchanged by matter and radiation.
Energy of one quantum is E = hf, where h is Planck's constant and f is frequency.
Planck's constant is h = 6.63 × 10^-34 J s.
For electromagnetic radiation, E = hc/λ.
Higher frequency means greater energy per photon.
A blackbody is an ideal body that absorbs all incident radiation and emits a continuous spectrum.
Blackbody radiation has a continuous spectrum, not a line spectrum.
The quantum theory states that energy exchange does not occur continuously, but in discrete bundles.

Electromagnetic Radiation and the Photon Model

Light has both wave and particle properties. Maxwell regarded light as an electromagnetic wave. The wave model explains interference, diffraction and polarization, while the photon model explains the photoelectric effect and Compton effect.

A photon is a packet of electromagnetic energy. In vacuum, all photons travel with speed c, although their energies and momenta may be different.

Speed of light in vacuum is c = 3.0 × 10^8 m s^-1.
Photon energy is E = hf = hc/λ.
Photon momentum is p = E/c = h/λ.
A photon has zero rest mass and zero electric charge.
A photon is not at rest. Its rest mass is zero, but it carries energy and momentum.
Shorter wavelength radiation has higher frequency and higher photon energy.
The energy order is γ-rays, X-rays, ultraviolet, visible, infrared, microwaves and radio waves.
Maxwell's theory describes light as electromagnetic waves, while the photon model describes light as particles.

Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of suitable frequency falls on it. The emitted electrons are called photoelectrons. The effect supports the particle nature of light.

Each emitted electron receives energy from one photon. Part of this energy is used to remove the electron from the metal surface. The remaining energy appears as the maximum kinetic energy of the electron.

The minimum energy needed to remove an electron from a metal is called its work function, represented by φ or W0.
Einstein's photoelectric equation is hf = φ + Kmax.
Maximum kinetic energy is Kmax = hf − φ.
Using stopping potential Vs, Kmax = eVs, so eVs = hf − φ.
Threshold frequency f0 is the minimum frequency below which no photoelectrons are emitted.
Work function is related to threshold frequency by φ = hf0.
Threshold wavelength is the longest wavelength that can produce photoelectric emission, λ0 = hc/φ.
For a work function of 4 eV, the threshold wavelength is approximately 309 nm.
The stopping potential depends on the frequency of incident light and the work function of the metal.
If frequency is below the threshold frequency, increasing intensity does not produce photoelectrons.

Laws and Observations of Photoelectric Emission

Photoelectric emission begins without an appreciable time delay when the incident frequency is above the threshold frequency. This immediate emission is explained by the one photon to one electron interaction.

The number of emitted electrons depends mainly on the intensity of incident radiation. The maximum kinetic energy depends on frequency, not intensity.

For light above threshold frequency, increasing intensity increases the photoelectric current.
If the intensity is doubled, the maximum number of emitted electrons becomes double, provided other conditions remain constant.
For fixed frequency above threshold, photoelectric current increases with intensity until saturation current is reached.
Saturation current is the maximum current obtained when all emitted electrons are collected.
For fixed intensity, increasing frequency increases the stopping potential and maximum kinetic energy.
Stopping potential is independent of intensity for a fixed frequency and metal.
The graph of maximum kinetic energy or stopping potential against frequency is a straight line above threshold frequency.
The graph of photoelectric current against frequency is zero below threshold frequency. Above threshold, emission occurs and the current depends on the intensity and collection conditions.
Photoelectric effect proves the particle nature of light because energy is transferred in individual photon electron interactions.

Intensity, Distance and Photocells

A photocell converts light energy into electrical energy by using photoelectric emission. For a point source, light intensity decreases according to the inverse square law. This changes the number of emitted electrons and hence the photocurrent.

When a source is moved farther from a photocell, its frequency remains unchanged. Therefore, the energy of each photon remains unchanged, but fewer photons reach the photocell per unit time.

For a point source, intensity I is proportional to 1/r^2.
If distance changes from 25 cm to 1 m, the distance becomes four times greater.
At four times the distance, intensity becomes 1/16 of its original value.
At constant potential difference, the number of emitted electrons becomes 1/16 as numerous in the 25 cm to 1 m example.
Changing intensity changes the number of photoelectrons, not their maximum kinetic energy.
A lens with the same diameter collects approximately the same total light from the Sun, even if its focal length changes.
Changing focal length changes the size and brightness of the formed image, but the total collected power can remain the same.
Thus, if the complete solar image is received by the photocell, replacing a 30 cm focal length lens with a 15 cm lens of the same diameter gives the same photocurrent I.

X-rays and the Reverse Photoelectric Process

X-rays are high-frequency electromagnetic radiation with very short wavelengths. They are produced when high-speed electrons strike a metal target and lose energy rapidly. In the traditional FSc treatment, X-ray production is described as the reverse process of the photoelectric effect.

The energy of an X-ray photon is calculated using E = hc/λ. Shorter wavelength X-rays have greater photon energy.

X-rays are electromagnetic waves and have zero rest mass and zero charge.
X-ray photons are produced when fast electrons are suddenly decelerated in a target.
The minimum wavelength of X-rays depends on the maximum energy of the bombarding electrons.
For λ = 1.0 × 10^-10 m, E = hc/λ ≈ 1.99 × 10^-15 J.
X-rays have higher energy and shorter wavelength than ultraviolet radiation.
The reverse process of photoelectric effect is commonly given in FSc texts as production of X-rays.
X-rays can show diffraction, confirming their wave nature.
X-ray photons also behave as particles because they carry definite energy and momentum.

Compton Effect and Pair Production

The Compton effect is the increase in wavelength of an X-ray or γ-ray photon after it collides with a loosely bound electron. The photon transfers some energy and momentum to the electron.

Compton scattering provides evidence for the particle nature of light. Pair production is another particle interaction in which a high-energy photon changes into matter particles near a nucleus.

In the Compton effect, the scattered photon has lower energy and longer wavelength than the incident photon.
Compton's experiment provides evidence for the particle nature of light.
Compton scattering does not prove only the wave nature of light.
Pair production is the conversion of a high-energy photon into an electron and a positron.
A γ-ray photon is required for pair production.
The minimum energy required is 1.02 MeV, equal to the combined rest energies of an electron and a positron.
The photon must usually interact near a nucleus so that momentum can be conserved.
For photon energies greater than 1.02 MeV, the probability of pair production increases as energy increases.
In pair production, electric charge is conserved because the electron has charge −e and the positron has charge +e.

de Broglie Matter Waves

De Broglie proposed that every moving particle has an associated wave. This is called the wave nature of matter. The wavelength is called the de Broglie wavelength.

The wavelength becomes smaller when momentum becomes larger. This is why wave effects are not normally noticed for large everyday objects.

De Broglie wavelength is λ = h/p.
For a non-relativistic particle, p = mv, so λ = h/mv.
For particles having the same velocity, the particle with greater mass has the shorter wavelength.
For an electron, proton, neutron and alpha particle moving with the same velocity, the alpha particle has the shortest wavelength because it has the greatest mass.
Momentum and de Broglie wavelength are inversely proportional.
The graph of p against λ is a decreasing rectangular hyperbola because p = h/λ.
The wave nature of matter is important for microscopic particles such as electrons.
The wave nature of matter does not mean that a particle loses its particle properties.

Special Relativity and Mass Energy Equivalence

Special relativity applies to objects moving at speeds comparable to the speed of light. It is based on the constancy of the speed of light and the equivalence of physical laws in all inertial frames.

In space time, three spatial dimensions are joined with time. Therefore, time is treated as the fourth dimension in addition to length, breadth and height.

The speed of light in vacuum is the same for all inertial observers.
No object having rest mass can reach or exceed the speed of light.
Relativistic mass is sometimes written as m = m0/√(1 − v^2/c^2).
As the speed of an object approaches c, its relativistic mass tends towards infinity.
An object moving exactly at the speed of light would require infinite energy if it had rest mass.
Relativistic energy is equivalent to relativistic mass by E = mc^2, or m = E/c^2.
The rest energy of an object is E0 = m0c^2.
Mass is not doubled at the speed of light. In the older relativistic-mass description, it becomes double at v approximately 0.866c, not at c.
The fourth dimension added to the three Cartesian dimensions is time.

Radiation Pressure and Photon Momentum

Photons transfer momentum when they are absorbed or reflected. This transfer produces radiation pressure and a force on the surface. The force depends on the intensity and the area receiving the radiation.

For a perfectly absorbing surface, each photon transfers its momentum once. For a perfectly reflecting surface, the momentum change is approximately twice as large.

Momentum of one photon is p = E/c.
For a perfectly absorbing surface, force F = IA/c.
For a perfectly reflecting surface, force F = 2IA/c.
Here I is intensity and A is the area receiving radiation.
For a circular absorber, A = πr^2.
If I = 1.4 × 10^3 W m^-2 and r = 2 m, a circular perfectly absorbing surface experiences F = IA/c ≈ 5.9 × 10^-5 N.
If the stated area is a spherical surface, A = 4πr^2 and the corresponding value is approximately 2.35 × 10^-4 N.
Radiation pressure on a perfectly absorbing surface is I/c, while on a perfectly reflecting surface it is 2I/c.

Key terms

Quantum: A discrete packet of energy exchanged by matter and radiation.
Photon: A packet of electromagnetic radiation having energy E = hf and momentum p = h/λ.
Blackbody: An ideal body that absorbs all incident radiation and emits a continuous spectrum.
Work function: The minimum energy required to remove an electron from a metal surface.
Photoelectric effect: The emission of electrons from a metal surface when suitable radiation falls on it.
Photoelectron: An electron emitted from a metal by the photoelectric effect.
Threshold frequency: The minimum frequency required to cause photoelectric emission from a particular metal.
Stopping potential: The reverse potential needed to reduce the maximum photoelectric current to zero.
Saturation current: The maximum photocurrent obtained when all emitted electrons are collected.
X-rays: High-frequency electromagnetic radiation produced by the sudden deceleration of fast electrons.
Compton effect: The increase in wavelength of a photon after scattering from an electron.
Pair production: The conversion of a high-energy photon into an electron and a positron.
de Broglie wavelength: The wavelength associated with a moving particle, given by λ = h/p.
Relativity: The theory describing space, time, motion and energy at speeds comparable to the speed of light.
Relativistic mass: The speed-dependent mass used in older treatments, which increases as speed approaches c.
Rest energy: The energy equivalent of an object's rest mass, given by E0 = m0c^2.
Radiation pressure: Pressure produced when electromagnetic radiation transfers momentum to a surface.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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