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 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.

Problem of Gradients: Equipotential lines change by 2 v m -1 in the x direction and 1v m -1 in the y direction. The potential does not change in z direction. Find the electric field E

Electric Filed Superposition of Electric Field

Electric Field The Superposition Principle for charge continuously distributed over an object line charge Surface Volume where: is radius position of point where the electric field is defined with respect to small volume dV with volume charge density V is volume of the charged object

Electric Field Electric field produced by uniformly charged ring at its axis with magnitude where: is radius position of point P where the electric field is defined with respect to center of the ring R is radius of ring Q is total charge of the ring

Electric Field

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

Volume with Sink To imagine what a sink would 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.

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:

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

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

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 curl of the field vectors.

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

Curl of Fields Curl B = μ  E + μ  ∂ E ∂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.

Faraday Cage 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 Disc First Dynamo 1831 First dynamo ever invented, described by Faraday in Experimental Researches in 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

Faraday Disc Generator B Magnetic filed varies with time and is given by B = B 0 sin  t, where B 0 is maximum Flux density

Example 1 Faraday Law A time-varying magnetic filed is given by:A time-varying magnetic filed is given by: Where B 0 is constant. It is desired to find the induced emf around the rectangular loop C in the xz- plane bounded by the lines x=0, x = z, z=0, and z = b. as shown in figure: x z b a O Choosing dS = dx.dza y according to the right hand rule the Total flux enclosed by the loop is therefore:

Example 1 ….. abB 0 abB 0   emf

Example 2 Faraday Law dS y x A rectangular loop of a wire with three sides fixed and the forth side movable is situated in a plane perpendicular to a uniform magnetic field B=B 0 a z as shown in the figure. Moving side is a conducting bar moving at the velocity of v 0 in y- direction. It is desired to find emf. generated around the loop C. The position of the bar at any time t is y 0 +v 0 t and considering dS=dx.dy a z v0ayv0ay C

Example 2 ….. dS y x v0ayv0ay Note that once the bar is moving to the right, the induced e.m.f. is negative and produces a current in the sense opposite to that of C. This polarity of the current is such that it gives rise to a magnetic field directed out of the paper. C