sphericity

Examples of sphericity in the following topics:

• Spherical and Plane Waves

  • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
  • In 1678, he proposed that every point that a luminous disturbance touches becomes itself a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.
  • Since the waves all come from one point source, the waves happen in a spherical pattern.
  • All the waves come from a single point source and are spherical .
  • When waves are produced from a point source, they are spherical waves.

• Triple Integrals in Spherical Coordinates

  • When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
  • When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
  • It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
  • Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
  • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

• Cylindrical and Spherical Coordinates

  • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.
  • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
  • Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.

• Gravitational Attraction of Spherical Bodies: A Uniform Sphere

  • The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center.
  • For highly symmetric shapes such as spheres or spherical shells, finding this point is simple.
  • A spherically symmetric object affects other objects gravitationally as if all of its mass were concentrated at its center,
  • If the object is a spherically symmetric shell (i.e., a hollow ball) then the net gravitational force on a body inside of it is zero.
  • That is, a mass $m$ within a spherically symmetric shell of mass $M$, will feel no net force (Statement 2 of Shell Theorem).

• Three-Dimensional Coordinate Systems

  • Often, you will need to be able to convert from spherical to Cartesian, or the other way around.
  • The spherical system is used commonly in mathematics and physics and has variables of $r$, $\theta$, and $\varphi$.
  • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.

• Introduction to Spherical and Cylindrical Harmonics

  • In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
  • Spherical coordinates are important when treating problems with spherical or nearly-spherical symmetry.
  • To a first approximation the earth is spherical and so is the hydrogen atom, with lots of other examples in-between.
  • On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
  • Obviously this would be much simpler if we used spherical coordinates, since then we could specify boundary conditions on, for example, the surface $x = r \cos \phi \sin \theta$ constant.

• Spherical Distribution of Charge

  • The charge distribution around a molecule is spherical in nature, and creates a sort of electrostatic "cloud" around the molecule.
  • This distribution around a charged molecule is spherical in nature, and creates a sort of electrostatic "cloud" around the molecule.
  • The attraction or repulsion forces within the spherical distribution of charge is stronger closer to the molecule, and becomes weaker as the distance from the molecule increases.
  • Describe shape of a Coulomb force from a spherical distribution of charge

• Newton's Rings

  • Newton's rings are a series of concentric circles centered at the point of contact between a spherical and a flat surface.
  • In 1717, Isaac Newton first analyzed an interference pattern caused by the reflection of light between a spherical surface and an adjacent flat surface.
  • Newton's rings appear as a series of concentric circles centered at the point of contact between the spherical and flat surfaces.
  • A spherical lens is placed on top of a flat glass surface.

• Image Formation by Spherical Mirrors: Reflection and Sign Conventions

  • This section will cover spherical mirrors.
  • Spherical mirrors can be either concave or convex.

• Boiling & Melting Points

  • The examples given in the first two rows are similar in that the molecules or atoms are spherical in shape and do not have permanent dipoles.
  • The upper row consists of roughly spherical molecules, whereas the isomers in the lower row have cylindrical or linear shaped molecules.
  • Spherically shaped molecules generally have relatively high melting points, which in some cases approach the boiling point.
  • The last compound, an isomer of octane, is nearly spherical and has an exceptionally high melting point (only 6º below the boiling point).