Astrophysics (Part III) - Light & Matter (Part I) Notes

Momentum and Energy Conservation

• Momentum and energy are conserved quantities in physics.
• Conservation laws dictate that these quantities cannot be created or destroyed, only transferred or transformed.

Angular Momentum

• Angular momentum is the tendency of spinning and rotating objects to maintain their circular motion.
• It is a conserved quantity in an isolated system, meaning it remains constant.
• $L = mvr$ (angular momentum), where:
  • $L$ is angular momentum
  • $m$ is mass
  • $v$ is velocity
  • $r$ is radius
• If one term in the equation decreases, another must increase to compensate.

Angular Momentum Example

• Ice skaters spin faster when arms are tucked in (radius decreases) and slower when arms are extended (radius increases) to conserve angular momentum.

Tidal Braking

• As Earth's spin decreases due to tidal braking, its angular momentum decreases.
• To conserve momentum, the moon's orbital radius increases by about 40 mm each year.

Orbits

• Based on Newton's 3rd Law, if a star exerts a gravitational force on a planet, the planet exerts an equal and opposite force back on the star.
• Both objects orbit their center of mass, or barycenter.
• The barycenter is the pivot point around which objects spin.
• Examples include: Star and Planet, Planet and Moon, or Binary Star Systems.

Energy

• Energy, like momentum, is a conserved quantity.
• Energy can't be created or destroyed, only transferred or transformed.
• Three main types of energy in physics:
  • Radiant (or radiative) Energy: energy of light.
  • Kinetic Energy: energy of a moving object (Thermal energy is a type of kinetic energy!).
  • Potential Energy: energy stored by virtue of an object's position.

Examples of Potential Energy

• Electric potential energy: energy possessed by an electric charge in an electric field.
• Chemical potential energy: energy of an atom due to its position within a molecule.
• Gravitational potential energy: energy of a mass within a gravitational field.

Conservation of Energy

• Objects at higher positions have more gravitational potential energy because they can fall for a longer time.
• When an object falls, it loses gravitational potential energy, which is converted into kinetic energy, causing the object to speed up.

Energy Transformations

• Examples of energy transformations:
  • Lifting a weight: chemical potential energy to gravitational potential energy.
  • Sliding into base: kinetic energy to thermal energy.
  • Campfire: chemical potential energy to thermal and radiant energy.
  • Diver: elastic potential to kinetic to gravitational potential to kinetic energy.

Mass vs. Energy

• Modern physics regards mass as another form of potential energy.
• Stars convert mass-energy into radiant and thermal energy.
• Energy is conserved: The sun loses $4.2$ billion kg of mass every second due to emitting light.

The Nature of Light

• Light is an oscillating electromagnetic field that doesn't require a physical material to propagate, unlike sound or earthquakes.
• Light also exhibits particle-like properties.

Anatomy of a Wave

• Wavelength ($\lambda$): distance spanned by one full cycle of the wave motion, measured in meters (m).
• Period (T): time for one complete cycle.
• Frequency (f): inverse of the period, measured in Hertz (Hz = $1/second$).
• $v = \lambda f$, where:
  • $v$ is speed
  • $\lambda$ is wavelength
  • $f$ is frequency

Light Waves

• All light waves travel at the speed of light in vacuum, denoted as $c$.
• $c = 3.0 \times 10^8 m/s$ (speed of light in vacuum).
• $v = \lambda f$

The Electromagnetic Spectrum

• $E_{photon} = hf$, where:
  • $E_{photon}$ is the energy of a photon
  • $h$ is Planck's constant ($h = 6.63 \times 10^{-34} J s$)
  • $f$ is frequency

Frequency and color

• Frequency determines color: blue light has a higher frequency and red light has a lower frequency.

Interactions Between Light and Matter

• Light and matter interact in four primary ways:
  1. Emission: light produced by matter.
     • Hot objects transform thermal energy into radiant energy.
     • Glow sticks and some insects convert chemical potential energy into radiant energy.
  2. Absorption: opaque matter absorbs light, converting radiant energy into thermal energy, causing the object to heat up. Plants convert radiant energy into chemical potential energy.
  3. Reflection / scattering: light bounces off matter.
     • Reflection: light bounces in a predictable direction from smooth surfaces.
     • Scattering: light bounces in all possible directions from rough surfaces.
  4. Transmission: light passes through transparent matter.
     • The path of light is bent, and its speed is decreased.

The Index of Refraction

• The speed of light in a material is characterized by its index of refraction.
• Light travels at speed $c$ in a vacuum but slows down in transparent objects like water or glass.

Faster Than Light

• It is possible to go faster than the speed of light, as long as the light is not in a vacuum.
• Cherenkov radiation: a shock wave of light emitted when a charged particle moves faster than the speed of light in a material (analogous to a sonic boom).

Concept Check

• Wave speed is given by $v = \lambda f$.
• When light slows down in a non-vacuum, its wavelength changes, not its frequency.
• If frequency changed, the light would gain or lose energy, violating the law of conservation of energy as given by $E_{photon} = hf$.