1 / 19

Gradient of Scalar Field

Gradient of Scalar Field. In Cartesian co-ordinates:. Gradient of Scalar Field. Gradient of Vector Field. Divergence of a vector field A is defined as the net outward flux of A per unit volume as the volume about the point tends to zero, i. Gradient of Vector Field.

amena-mayer
Download Presentation

Gradient of Scalar Field

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. Gradient of Scalar Field In Cartesian co-ordinates:

  2. Gradient of Scalar Field

  3. Gradient of Vector Field Divergence of a vector field Ais defined as the net outward flux of A per unit volume as the volume about the point tends to zero, i.

  4. Gradient of Vector Field The flux of a vector field (the integral term in this equation) is analogous to the total flow of an incompressible fluid through a surface (S in the equation). For a closed surface, the total flow through the surface is Often zero - if A represents water flow, for example, then the total amount of water flowing into a surface is always matched by the amount coming out.

  5. Problem of Gradients: Equipotential lines change by 2 v m-1in the x direction and 1v m-1 in the y direction. The potential does not change in z direction. Find the electric field E ^ ^ E = - V = - (x ∂V/∂x + y∂V/ ∂y) = x 2 + y 1 = 2.24  26.6  Vm-1

  6. Electric Filed Superposition 1 Q1 Q2 Q3 Vp = ( ) + + 4 0 r1 r2 r3 1 3 Qn  VP = 4 0 rn N=1

  7. Electric Field Superposition L 1 VL = dL  (V) line charge r 4 0 s 1 Surface   dS VL = 4 0 r v 1   dv VL = Volume 4 0 r

  8. Electric Field

  9. Volume with Source Source:-Divergence is Non-Zero at the lit point.

  10. Volume with Sink To imagine what a sinkwould look Like; Imagine video-taping the sparkler and replaying it backwards. In that case the sparks would flow back into the lit point – this would then appear to be a sink.

  11. Divergence The net outward flow per unit volume is therefore a measure of the strength of the enclosed source (note this is a scalar quantity). In Cartesian co-ordinates the expression for div A is

  12. Divergence of Magnetic Filed The interpretation of this zero divergence relation is that no sources or sinks of magnetic charge exist.  • B = 0

  13. Divergence of Electric Filed Electric Field The electric flux density D, it can be shown that Where D is Flux Density ,  v is the free electric charge density  • D =  v

  14. Divergence & Curl of Fields visualize the fields as resulting from forces generated by whirlpool-like vortices in the ether. Maxwell was therefore interested in the rotation (or circulation, or “vorticity”) of the fields which today we represent in terms of the curlof the field vectors.

  15. Curl of Fields Curl E = ∂ B/ ∂r which describes the generation of a “vortex” in the electric field resulting from a time varying magnetic field.

  16. Curl of Fields ∂ E Curl B = μE + μ ∂t µóE term describes the generation of a vortex in the magnetic field resulting from the passage of an electric current through the medium. The second term represents a further contribution to the vortex resulting from a time-varying electric field.

  17. A human-size Faraday cage which allows someone to stand inside while the discharge from a Tesla coil is directed towards it. The person inside is asked to hold a fluorescent tube, which does not light. While similar tubes balanced against the sides of the cage do light when the discharge is enabled. Faraday Cage

  18. Faraday Disc First Dynamo 1831 First dynamo ever invented, described by Faraday in Experimental Researches in 1831. The device consists of a copper disc that can be rotated in the narrow gap between the poles of a magnet. Contacts near the centre and edge of the disc (made more conducting by the application of mercury) develop a voltage between them

  19. Faraday Disc Generator B Magnetic filed varies with time and is given by B = B0 sint, where B0 is maximum Flux density

More Related