Physics Notes - Principles of Modern Physics PDF

## Dear Dr. Rand ### Chapter 1: Principles of Modern Physics **What we learn from reviewing the applications of electromagnetic radiation:** * **The nature of electromagnetic radiation** * **Its properties** * **The mechanism of its generation** **Light is part of electromagnetic radiation** Th...

## Dear Dr. Rand ### Chapter 1: Principles of Modern Physics **What we learn from reviewing the applications of electromagnetic radiation:** * **The nature of electromagnetic radiation** * **Its properties** * **The mechanism of its generation** **Light is part of electromagnetic radiation** This chapter will cover the nature of electromagnetic waves, no matter what type they are. In the next chapter, we will discuss the mechanism of their generation and emission at the atomic and molecular levels. We will focus on the visible part of electromagnetic radiation in the third chapter due to its importance in understanding these waves. **The dual nature of electromagnetic radiation as waves and particles:** Two distinct topics are covered in any textbook on traditional physics: particle and wave. - **Particle:** * Localized in a specific position at a given time * Has: * Mass * Velocity (if it moves) * Potential or kinetic energy, or both, depending on the state of the particle. - **Wave:** * Spreads out, meaning it is not localized in a particular point at a given moment. * Known for its: * Frequency * Wavelength * Energy spread across space and not limited to a specific position. **Interference and diffraction are the two phenomena distinguishing waves from particles.** **Electromagnetic radiation can be represented using an electromagnetic wave that propagates in a straight line through a vacuum at a speed of:** - **$c=300000Km.s^{-1}$** - **$c=3\times10^{8} m.s^{-1}$** **Explanation:** Imagine two things in our lives: objects or particles (smaller objects like balls, our body, a wall, an electron, etc.)... ## Understanding the Nature of Light * Young's experiment was a double-slit experiment that showed that light has wave properties. * Fraunhofer's experiment in diffraction showed that light is also wave-like. * Maxwell's experiment in 1865 showed that the wave nature of electromagnetic radiation is general. * Maxwell's experiment also showed that electromagnetic waves emitted when charged particles like electrons accelerate. * Plank suggested that the energy of electromagnetic radiation is quantized to explain the radiation of a black body. The energy was distributed across different wavelengths. * Each of these quanta was called *photon*. * Plank concluded that: * photons behave like material particles when interacting with electrons. * The energy of photons is a quantity related to the frequency of the vibrating charge (of the emitted radiation) or the frequency of radiation: * $E_{min} = hf$ * Where $h = 6.626 \times 10^{-34} jouls$ is Planck's constant. * These charged vibrating quantities have energy that is an integer multiple of $hf$: * $E = nhf$ * This is the *Planck's quantum hypothesis*. * It means that vibration energy can only have specific values such as: * $1hf, 2hf, h, 3hf…$ * This means that vibration energy is not continuous but is quantized, meaning it is composed only of discrete quanta with the smallest amount being $hf$. ## Ordering Electromagnetic Radiation (important) **Ordering based on decreasing wavelength (increasing energy and frequency): ** * Gamma rays * X-rays * Ultraviolet rays * Visible light * Infrared rays * Microwaves * Radio waves **Notes:** * Radio waves have a relatively large wavelength. * X-rays and gamma rays have very small wavelengths. * Visible light is part of the electromagnetic spectrum, but it has a different wavelength and frequency. **The figure shows:** * Frequency increases on the y-axis. * Wavelength increases on the x-axis. ## Units | Symbol | Value relative to the meter | |---|---| | T | 10¹² | | G | 10⁹ | | M | 10⁶ | | k | 10³ | | h | 10² | | da | 10¹ | | d | 10⁻¹ | | c | 10⁻² | | m | 10⁻³ | | μ | 10⁻⁶ | | n | 10⁻⁹ | | A | 10⁻¹⁰ | | p | 10⁻¹² | | f | 10⁻¹⁵ | **Notes**: * The wavelength of 400 nm (4 x 10⁻⁷ m) is known as the visible range. * The ratio between wavelength and frequency is inverse: $λ = \frac{c}{f}$. * The ratio between energy and frequency is direct: $E = hf$. * It follows that frequency and energy are on the same team, while wavelength is on its own. **The following table is very important and will stay with us throughout physics. It is wise to commit it to memory.** **Einstein's explanation of the photoelectric effect:** Einstein proposed in 1905 an explanation for the photoelectric effect based on: * Planck’s hypothesis. * The interaction of photons with electrons behaves like collisions between particles. **Notes:** * It is important to know that the photoelectric effect involves a collision between a photon and an electron in an atom. * If the energy of this photon is greater than or equal to the ionization energy, the electron will be released. * The higher the frequency of the photon, the greater its energy and the greater its ability to release the electron. **The smallest value of the frequency, necessary to release an electron, is called the cut-off frequency. The energy of the corresponding photon is called the work function.** * The energy of the photon, according to Einstein, is: * **$E = hf = hc/λ$** * Using this equation, we can express the energy of a photon in electron volts: * **$E = 12400/λ $** **Where:** * **$λ$ **is expressed in *angstrom*. * **$E$ **is expressed in *electron volts*. **Notes**: * If the radiation is monochromatic, it will consist of photons with equal energy. The intensity is proportional to the number of photons. * In the case of continuous radiation, as in white light, it consists of photons with different energies based on the different wavelengths which have equal numbers of particles. The photon theory provides a clear explanation for the photoelectric effect. An electron absorbs a complete photon. If the energy of the photon is greater than or equal to the ionization energy, it will release the electron. * The higher the frequency of the photon, the greater its energy and the greater its ability to release the electron. ## The Compton Effect * The Compton effect has provided new evidence for the particle nature of light, the photon theory. * This prompted scientists to conduct various experiments, one of which was the Compton scattering of X-rays. * Compton observed that when X-rays are scattered by electrons, the scattered X-rays have a slightly lower frequency and hence larger wavelength than the incident X-rays. * Classical wave theory could not explain this phenomenon. However, Compton was able to explain the observed frequency shift by assuming that the X-rays are particles (photons). * The interaction can then be viewed like a collision between a photon and a free electron. **Notes:** * Compton assumed that the photon in this case is a particle with energy $E$ and momentum $p$. * The equations for energy and momentum are then given by: * **$E = hf = hc/λ$** * **$p = hf/c = h/λ$** **The first equation is familiar while the second equation is derived from the wave equation for light particles** * Since a photon is a particle, its momentum can be expressed in terms of mass and velocity: **$p=mc$** * Substituting in the momentum equation, we find the **effective mass of a photon** to be: **$ m = h/λc$ ** **This is called the Compton effect.** **Notes:** * It is important to realize that a photon has no rest mass. ## The Compton Effect: results, and equations * The Compton effect, along with the photoelectric effect, supports the particle nature of electromagnetic radiation (photon theory). * Compton scattered short-wavelength light (X-rays) from different materials in 1923 and found that: * The scattered light’s wavelength is a bit longer than the incident wavelength and, therefore, its frequency is lower. * The scattered light’s energy is lower. * This can be interpreted using the photon model: a collision occurs between a photon and an electron of an atom. * Using the conservation laws of momentum and energy, Compton derived the following equation for the scattered photon's wavelength: **$λ' = λ + (1-cosθ) \frac{h}{m_oc}$** **Where:** * $λ' = λ$ wavelength of the scattered photon. * $λ'$ wavelength of the incident photon. * $m_oc = h$ Compton wavelength. **Notes:** * Notice that the wavelength of the scattered photon depends on $θ$ and is known. * The incident photons' wavelength is also known. * Therefore, the only unknown in this formula is the scattered photon's wavelength. **Explanation:** * The Compton effect occurs when a high-energy photon (e.g., X-ray) interacts with an electron bound to an atom. * The photon can transfer some of its energy to the electron, causing a change in direction (scattering). * This energy transfer results in a decrease in the photon's energy, which is equivalent to an increase in its wavelength. ### The Compton effect: example * A photon of wavelength $0.140 nm$ is scattered via the Compton Effect from a thin carbon sheet. Calculate the wavelength of the scattered photon at angles $0^\circ$ , $90^\circ$ , $180^\circ$ **Solution:** * The problem is an example of the Compton effect. * $λ' = λ + (1-cosθ) \frac{h}{m_oc}$ * ** At the angle 0 = 0: ** * **$ cos θ = 1 $ ** * **(1 - cos θ) = 0 ** * **λ' = λ = 0.140 nm** * ** At the angle 0 = 90: ** * **cos θ = 0** * **(1 - cos θ) = 1 ** * **$λ' = λ + \frac{h}{m_oc}$** * **At the angle 0 = 180: ** * **cos θ = -1** * **(1 - cos θ) = 2** * **$λ' = λ + 2\frac{h}{m_oc}$** **Notes:** * Notice that the maximum shift in wavelength occurs in the case of backscattering (θ = 180°). * It is twice the Compton wavelength **Conclusion:** The Compton effect shows that photons behave like particles, with momentum and energy. It confirms de Broglie's wave-particle duality hypothesis. ## Wave-particle duality of matter Louis de Broglie, a French physicist, proposed that matter can also have wave properties in addition to its particulate nature. This is the *wave-particle duality of matter*. He proposed a relationship between the momentum of a particle and its wavelength, which is now known as the de Broglie wavelength. * de Broglie's suggestion was in contrast to prevailing concepts. At the time, it had no experimental support. * This idea gained prominence three years later, and de Broglie received the Nobel Prize in 1929 for discovering the wave nature of electrons. De Broglie's suggestion was: * If radiation with a definite wavelength has a linear momentum $p$. * A particle with a defined momentum *p* should have a characteristic wavelength $λ$. * This equation is now known as de Broglie wavelength: **$λ = \frac{h}{p} = \frac{h}{mv}$** * Where: * $λ$ is the wavelength of the particle. * $h$ is Planck's constant. * *m* is the mass of the particle. * *v* is the speed of the particle. This equation suggests that particles, like electrons, can also exhibit wavelike properties. **The smaller the speed of an electron, the larger the wavelength of the electron wave.** **Experiment by Davisson and Germer (1927) in the United States:** * The experiment confirmed de Broglie's hypothesis. * It proved that electrons behave like waves. * A beam of electrons was directed at a nickel crystal, and a diffraction pattern was observed. * The angle of diffraction and the wavelength of the electron were consistent with de Broglie's theory. * **Calculating the de Broglie wavelength for an electron travelling at 5.9 x 10^6 m/s:** * $λ = h/ mv = 6.626 x 10^{-34} / 9.11 × 10^{-31} × 5.9 × 10^6 = 1 × 10^{-10} $ m * **Why is this wavelength measurable?** * Because the distance between atoms in crystals is of the order of an angstrom. * **Note:** Diffraction only occurs if the size of the slit or opening (the distance between the atoms) is of the order of the wavelength. The consequences of de Broglie's idea in electron microscopy: * This discovery led to the development of electron microscopes. * They have a higher magnification than light microscopes. * Their resolution power is much higher, allowing us to see things that were previously impossible to see with light microscopes. * Since the wavelength of electrons is shorter than that of visible light, electron microscopes can resolve features much smaller than those resolvable with light microscopes. * de Broglie’s equation is applied to a plane wave, which describes particles with an exactly determined momentum. * In reality, most particles have a distribution of momenta. * This distribution determines the de Broglie wave’s wavelength. * The relationship between the wavelength and momentum can be easily expressed by: * **2d sin(θ) = mλ… where m= 1, 2, 3**