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www.GOALias.blogspot.com1.15 APPLICATIONS OF GAUSS’S LAWThe electric field due to a general charge distribution is, as seen above,given by Eq. (1.27). In practice, except for some special cases, thesummation (or integration) involved in this equation cannot be carriedout to give electric field at every point inspace. For some symmetric chargeconfigurations, however, it is possible toobtain the electric field in a simple way usingthe Gauss’s law. This is best understood bysome examples.1.15.1 Field due to an infinitelylong straight uniformlycharged wireConsider an infinitely long thin straight wirewith uniform linear charge density λ. The wireis obviously an axis of symmetry. Suppose wetake the radial vector from O to P and rotate itaround the wire. The points P, P′, P′′ soobtained are completely equivalent withrespect to the charged wire. This implies thatthe electric field must have the same magnitudeat these points. The direction of electric field atevery point must be radial (outward if λ > 0,inward if λ < 0). This is clear from Fig. 1.29.Consider a pair of line elements P 1and P 2of the wire, as shown. The electric fieldsproduced by the two elements of the pair whensummed give a resultant electric field whichis radial (the components normal to the radialvector cancel). This is true for any such pairand hence the total field at any point P isradial. Finally, since the wire is infinite,electric field does not depend on the positionof P along the length of the wire. In short, theelectric field is everywhere radial in the planecutting the wire normally, and its magnitudedepends only on the radial distance r.To calculate the field, imagine a cylindricalGaussian surface, as shown in the Fig. 1.29(b).Since the field is everywhere radial, fluxthrough the two ends of the cylindricalGaussian surface is zero. At the cylindricalpart of the surface, E is normal to the surfaceat every point, and its magnitude is constant,since it depends only on r. The surface areaof the curved part is 2πrl, where l is the lengthof the cylinder.Electric Chargesand FieldsFIGURE 1.29 (a) Electric field due to aninfinitely long thin straight wire is radial,(b) The Gaussian surface for a long thinwire of uniform linear charge density.37