Provisional Presentation File 10Aug20111CMDAYS 2011 Aravamudhan The contents of the Slides #2-5, 8 & 9 are materials which have been presented on earlier occasions, and included here for a recapitulation. Download a summary of the results on induced field calculations for a magnetized material PowerPoint File for CMDAYS2011 oral presentation

R1R1 RnRn Radial Vector defined by a polar angle θ w.r.to Z Pol ar an gle R1R1 RnRn With “C= R i / r i, i=1, n” For a sphere of radius =0.25 units, and the polar angle changes at intervals of 2. 5˚ There will be 144 intervals. Circumference= 2π/4 so that the diameter of each sphere on the circumference = ; radius = C = R/r = 0.25 / = [ / ] = Log ( ) = (r/R) 3 = e-5 = Z-Axis; The Direction of magnetic field Equation for calculating the number of spheres, the dipole moments, along the radial vector is as given below: Using these equation ‘n’ along the vector length is calculated, for the direction with polar angle θ. Which is ‘σ’ per spherical magnetic moment x number of such spheres ‘n’. σ θ =σ x n. At the tip of the vector, there is circle along which magnetic moment have to be calculated. This circle has radius equal to ‘R sinθ’. The number of dipoles along the length of the circumference = 2 π R sinθ/2.r = π R/ r sinθ. Again, (R/ r) is a constant by earlier criteria. This circular base of the cone with apex angle equal to the polar angle θ, has radius equal to ‘R sin θ’: See Textbox below FIGURE-8 A two dimensional cross section for detailed perspective in the next slide 10Aug20112CMDAYS 2011 Aravamudhan

Line defined by Polar angle θ / direction of radial vector A dimensional consideration for the basis of the equations used. (in the next slide) 10Aug20113CMDAYS 2011 Aravamudhan

In the equation on the left if the polar angle θ is chosen to be zero, then the above depiction would result. 10Aug20114CMDAYS 2011 Aravamudhan

Volume of the sphere at distance ‘R’ is (4/3)π r 3 = V s with r = C(a constant) x R The induced field at this point will be = X v (V s )/R 3 where is the Volume susceptibility ( independent of the V s ). V s =Constant x R 3 so that the contribution from every one of the spheres has the same value F s = σ. This is a situation of discrete summing possibility. XvXv 10Aug20115CMDAYS 2011 Aravamudhan

The differential form for the above equation would be, for an infinitesimally small change in R, say dR, dF R = σ · dR, then for the length of the Vector, from R min to R max, F T = σ ·∫ dR R min R max The differential dR would have an expression in terms of the ‘dn’ where ‘dn’ is the change in the number of dipoles for an infinitesimal change in R, and this expression can be obtained from the equation for n, given in the previous slide in terms of the R. Also including the θ and φ dependence would call for defining the shape of the specimen which in turn would result in the procedure for evaluating integrals of the same complexity as encountered in earlier methods. Thus the simplification lies mainly because of the possibility of discrete summing. 10Aug20116CMDAYS 2011 Aravamudhan

Magnetic Field can be present even in vacuum; but it has to be appreciated that the Magnetic Dipole Moment (induced in particular) is a property of magnetized material. The spontaneous magnetizations in Magnetic materials would not be dealt with in this presentation. Then the following questions arise when subdivided magnetic moments are closely packed and each moment is compared with the moment of a bar magnet which is usually the comparisons while introducing magnetic moments. What in principle the acquired property when a magnetizable material is placed in a magnetic field? It is magnetic moment of the entire specimen which is evident. No specific evidence is available as to at which point this total magnetic moment should be located within the sample. When the magnetic field intensity is described by a Vector of certain magnitude and direction, it does not make it explicit where all in space this magnetic field can be effective. For the later perspective the lines of force has to be drawn which the magnetic field spans. Lines of force convention requires several lines to be placed adjacent, then if each line can be thought to be interacting with the next line- (“ the lines of force do not intersect “); The magnetic field in the same space from two different magnets would cause a resultant field? But the Field present in one place does not induce secondary field, and a magnetic moment induces secondary field. Should the Magnetization be, then, represented in terms of the mathematical vector maps? Magnetization is a Vector (but not a Field?) 10Aug20117CMDAYS 2011 Aravamudhan

10Aug2011CMDAYS 2011 Aravamudhan8 N S N S N Set of equally spaced parallel lines indicates homogeneous field No. of lines within a unit area of cross section [in plane perpendicular] Is the field strength/ Intensity of magnetic field Area of cross section Inhomogeneous Fields Presence of a Magnetic field is pictorially depicted by a vector-line pointing along the direction in which an isolated North pole would move After this illustration on magnetic Field, in the next slide, Magnetic Moments would be considered It is important to realize the difference between two poles forming magnet pole faces and two poles making up the dipole with a dipole moment

10Aug2011CMDAYS 2011 Aravamudhan9 N S ‘m’ is the pole strength, and ‘2l’ is the distance between the two poles Then the magnetic moment vector of magnitude μ is defined by: The moment has the direction from south to the north. Thus this sense of the vector seems to be opposite to what was said of direction of field ! N S N m m 2l N N S N + N An illustration of current flowing in a circular coil, direction of induced field at the centre and the secondary field distribution around and the movement of isolated north pole Primary moment at centre And Secondary field at points around the moment. And, the direction of movement of isolated north pole While considering a magnet,it is the field in the space between the North pole face and South pole face which is pertinent. There is not much concern about using the field values at distant points from the poles! No dipole is assigned nor any dipole-moment is recognizable for this configuration north & south poles. When a dipole is considered, its moment is placed in the region between the two poles forming the dipole. And the field at distant points from the dipole is what is pertinent. The next slide poses a question that arises in this context X X X X X X X N Lines of force between the two poles/ a visualization if there is no material between the two poles