cross section

Examples of cross section in the following topics:

• Absorption

  • Phenomenologically you can imagine that there are many independent absorbers in the beam, each with a cross section $\sigma_\nu$ and a number density $n$.
  • You can think of this as the cross section per unit mass of the absorbers.

• A Physical Aside: Einstein coefficients

  • Fermi's Golden Rule relates the cross-section for a process to a quantum mechanical matrix element and the phase space available for the products.
  • Because quantum mechanics for the most part is time reversible, the cross-section for the forward and reverse reactions are related.

• Driven Harmonic Oscillator

  • Let's imagine that our harmonic oscillator is driven by incoming electromagnetic wave.Using the assumptions from the section on scattering and the radiation reaction we have
  • Let's divide by the Poynting vector $\left \langle S \right \rangle = (c/8\pi) E_0^2$ to get the scattering cross-section
  • Near the resonance the cross-section has the same profile at the spontaneous emission.

• Flow Rate and the Equation of Continuity

  • where Q is the flow rate, v is the velocity of the fluid, and a is the area of the cross section of the space the fluid is moving through.
  • Speed increases when cross-sectional area decreases, and speed decreases when cross-sectional area increases.

• Volume

  • The volume of a cylinder: the cross-sectional area times the height of the cylinder.
  • Measuring cups, as seen in , work by taking a known cross sectional area of a cup and multiplying that by a variable height.
  • Since liquid will always cover the cross section (if there is enough liquid), adding more liquid will increase the height inside the container.

• Bound-Free Transitions and Milne Relations

  • Let's calculate the cross-section for a photon with $\bar{h}\omega \gg 13.6 Z^2$~eV to ionize a hydrogen-like ion from the ground state.
  • Using this in the formula for the differential cross-section and multiplying by $4\pi$ gives
  • The total cross-section for a photon of frequency $\omega$ to ionize an electron from a hydrogenic atom in state $n$ is
  • More interesting is the fact that you can relate the cross section for ionization to that of recombination, through the Milne relations.
  • Let $\sigma_{fb}(v)$ be the cross section for recombination for electrons with velocity $v$, then we have a recombination rate per unit volume of

• Pascal's Principle

  • As an example, referring to , a downwards force of 10 N is applied to a bottle filled with a static liquid of constant density ρ at the spout of cross-sectional area of 5 cm2, yielding an applied pressure of 2 N/cm2.
  • The cross-sectional area of the bottle changes with height so that at the bottom of the bottle the cross-sectional area is 500 cm2.
  • Since the cross-sectional area at the bottom of the bottle is 100 times larger than at the top, the force contributing to the pressure at the bottom of the bottle is 1000 N plus the force from the weight of the static fluid in the bottle.
  • A downwards force of 10 N is applied to a bottle filled with a static liquid of constant density ρ at the spout of cross-sectional area of 5 cm2, yielding an applied pressure of 2 N/cm2.

• Elasticity, Stress, and Strain

  • Additionally, the change in length is proportional to the original length L0 and inversely proportional to the cross-sectional area of the wire or rod.
  • For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.

• Energy Stored in a Magnetic Field

  • Let's consider Fig 1 , an example of a solenoid (ℓ: length, N: number of turns, I: current, A: cross-section area) that works as an inductor.
  • Magnetic field created by a solenoid (cross-sectional view) described using field lines.

• X-Ray Imaging and CT Scans

  • Tomography refers to imaging by sections, or sectioning. demonstrates this concept.
  • The three-dimensional image is broken down into sections.
  • (S1) shows a section from the left and (S2) shows a section from the right.
  • Doctors can cross reference the images with known properties of the same material and determine if there are any inconsistencies or problems.