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www.GOALias.blogspot.comPhysicsKARL FRIEDRICH GAUSS (1777 – 1855)Karl Friedrich Gauss(1777 – 1855) He was achild prodigy and was giftedin mathematics, physics,engineering, astronomyand even land surveying.The properties of numbersfascinated him, and in hiswork he anticipated majormathematical developmentof later times. Along withWilhelm Welser, he built thefirst electric telegraph in1833. His mathematicaltheory of curved surfacelaid the foundation for thelater work of Riemann.The situation is radically different for magnetic fieldswhich are continuous and form closed loops. Examine theGaussian surfaces represented by i or ii in Fig 5.3(a) orFig. 5.3(b). Both cases visually demonstrate that thenumber of magnetic field lines leaving the surface isbalanced by the number of lines entering it. The netmagnetic flux is zero for both the surfaces. This is truefor any closed surface.FIGURE 5.6Consider a small vector area element ∆S of a closedsurface S as in Fig. 5.6. The magnetic flux through ÄS isdefined as ∆φ B= B.∆S, where B is the field at ∆S. We divideS into many small area elements and calculate theindividual flux through each. Then, the net flux φ Bis,φB= ∑∆ φ = ∑Bg ∆ S = 0(5.9)B' all ' ' all 'where ‘all’ stands for ‘all area elements ∆S′. Compare thiswith the Gauss’s law of electrostatics. The flux through a closed surfacein that case is given byq∑ Eg∆S = ε0where q is the electric charge enclosed by the surface.The difference between the Gauss’s law of magnetism and that forelectrostatics is a reflection of the fact that isolated magnetic poles (alsocalled monopoles) are not known to exist. There are no sources or sinksof B; the simplest magnetic element is a dipole or a current loop. Allmagnetic phenomena can be explained in terms of an arrangement ofdipoles and/or current loops.Thus, Gauss’s law for magnetism is:The net magnetic flux through any closed surface is zero.182EXAMPLE 5.6Example 5.6 Many of the diagrams given in Fig. 5.7 show magneticfield lines (thick lines in the figure) wrongly. Point out what is wrongwith them. Some of them may describe electrostatic field lines correctly.Point out which ones.