Note for Electromagnetic Field - EMF By P Praveen Kumar
- Electromagnetic Field - EMF
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UNIT-I 1. State and Prove Gauss’s law. Express Gauss’s in both integral and differential forms Gauss's Law: Gauss's law is one of the fundamental laws of electromagnetism and it states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface. Fig 1: Gauss's Law Let us consider a point charge Q located in an isotropic homogeneous medium of dielectric constant flux density at a distance r on a surface enclosing the charge is given by . The ...............................................(1) If we consider an elementary area ds, the amount of flux passing through the elementary area is given by .....................................(2) But , is the elementary solid angle subtended by the area at the location of Q. Therefore we can write For a closed surface enclosing the charge, we can write 2. Write Applications of Gausses Law. Ans: Application of Gauss's Law: Gauss's law is particularly useful in computing or where the charge distribution has some symmetry. We shall illustrate the application of Gauss's Law with some examples.
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1. An infinite line charge As the first example of illustration of use of Gauss's law, let consider the problem of determination of the electric field produced by an infinite line charge of density LC/m. Let us consider a line charge positioned along the z-axis as shown in Fig. 2.4(a) (next slide). Since the line charge is assumed to be infinitely long, the electric field will be of the form as shown in Fig. 2.4(b) (next slide). If we consider a close cylindrical surface as shown in Fig. 2.4(a), using Gauss's theorm we can write, .....................................(2.15) Considering the fact that the unit normal vector to areas S1 and S3 are perpendicular to the electric field, the surface integrals for the top and bottom surfaces evaluates to zero. Hence we can write, Fig 2.4: Infinite Line Charge
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.....................................(2.16) 2. Infinite Sheet of Charge As a second example of application of Gauss's theorem, we consider an infinite charged sheet covering the x-z plane as shown in figure 2.5. Assuming a surface charge density of having sides for the infinite surface charge, if we consider a cylindrical volume placed symmetrically as shown in figure 5, we can write: ..............(2.17) Fig 2.5: Infinite Sheet of Charge It may be noted that the electric field strength is independent of distance. This is true for the infinite plane of charge; electric lines of force on either side of the charge will be perpendicular to the sheet and extend to infinity as parallel lines. As number of lines of force per unit area gives the strength of the field, the field becomes independent of distance. For a finite charge sheet, the field will be a function of distance. 3. Uniformly Charged Sphere
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Let us consider a sphere of radius r0 having a uniform volume charge density of v C/m3. To determine everywhere, inside and outside the sphere, we construct Gaussian surfaces of radius r < r0 and r > r0 as shown in Fig. 2.6 (a) and Fig. 2.6(b). For the region ; the total enclosed charge will be .........................(2.18) Fig 2.6: Uniformly Charged Sphere By applying Gauss's theorem, ...............(2.19) Therefore ...............................................(2.20) For the region ; the total enclosed charge will be ....................................................................(2.21) By applying Gauss's theorem, .....................................................(2.22)
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