1 / 7

Lecture 8: Div(ergence)

Lecture 8: Div(ergence). Consider vector field and. outside. Flux (c.f. fluid flow of Lecture 6) of out of S . inside. e.g, if is the fluid velocity (in m s -1 ), . S. rate of flow of material (in kg s -1 ) out of S

kynthia
Télécharger la présentation

Lecture 8: Div(ergence)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 8: Div(ergence) • Consider vector field and outside Flux (c.f. fluid flow of Lecture 6) of out of S inside e.g, if is the fluid velocity (in m s-1), S • rate of flow of material (in kg s-1) out of S • For many vector fields, e.g. incompressible fluid velocity fields, constant fields and magnetic fields Fluid density • But sometimes and we define div(ergence) for these cases by A scalar giving flux/unit volume (in s-1) out of

  2. In Cartesians • Consider tiny volume with sides dx, dy, dz so that varies linearly across it • Consider opposite areas 1 and 2 for just these two areas = dx Ax 1 dx 2 • Similar contributions from other pairs of surfaces

  3. Simple Examples

  4. Graphical Representation where brightness encodes value of scalar function

  5. Physics Examples • Incompressible ( = constant) fluid with velocity field because for any closed surface as much fluid flows out as flows in since for any closed surface • Constant field • Compressible (  constant) fluid:  can change if material flows out/in of dV mass leaving dV per unit time A continuity equation: “density drops if stuff flows out”

  6. Lines of Force • Sometimes represent vector fields by lines of force • Direction of lines is direction of • Density of lines measures • Note that div ( ) is the rate of creation of field lines, not whether they are diverging • If these field lines all close, then 2 1 1 • e.g. for magnetic field lines, there are no magnetic monopoles that could create field lines larger at 1 than 2 +Q • Electric field lines begin and end on charges: sources in dV yield positive divergence sinks in dV yield negative divergence -Q

  7. Further Physics Examples (s-1) charge density (C m-3)volume (m3) current density (A m-2) surface (m2) number density (m-3) of charges drift velocity field charge of current carrier • “Charge density drops if charges flow out of dV” (continuity equation) • Maxwell’s Equations link properties of particles (charge) to properties of fields • In quantum mechanics, similar continuity equations emerge from the Schrödinger Equation in which  becomes the probability density of a particle being in a given volume and becomes a probability current

More Related