"Explore practical applications of Gauss's law in electromagnetics with Dr. Erwin Sitompul at President University. Learn about electric flux density, divergence, and more through detailed examples. Follow along for in-depth insights into differential volume elements and surface integrals in electromagnetic field analysis."
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Engineering Electromagnetics Lecture 4 Dr.-Ing. Erwin Sitompul President University http://zitompul.wordpress.com 2 0 1 5 President University Erwin Sitompul EEM 4/1
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element We are now going to apply the methods of Gauss s law to a slightly different type of problem: a surface without symmetry. We have to choose such a very small closed surface that D is almost constant over the surface, and the small change in D may be adequately represented by using the first two terms of the Taylor s-series expansion for D. The result will become more nearly correct as the volume enclosed by the Gaussian surface decreases. We intend eventually to allow this volume to approach zero. President University Erwin Sitompul EEM 4/2
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Taylor s Series Expansion + = ( ) ( ) f x f x x 0 ( ) f x 0 = + x x x 0x 0 A point near x0 = + ( ) f x ( ) f x x f x 0 n ( ) 1! ( ) 2! ( ) ! n f x f x = + + + + 2 n ( ) f x ( ) f x ( ) ( ) 0 0 0 x x x 0 Only the linear terms are used for the linearization President University Erwin Sitompul EEM 4/3
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element Consider any point P, located by a rectangular coordinate system. The value of D at the point P may be expressed in rectangular components: D D = + + D a a a D 0 0 0 0 x x y y z z We now choose as our closed surface, the small rectangular box, centered at P, having sides of lengths x, y, and z, and apply Gauss s law: D d = + + = S d Q S left right bottom + + + D S front back top S President University Erwin Sitompul EEM 4/4
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element We will now consider the front surface in detail. The surface element is very small, thus D is essentially constant over this surface (a portion of the entire closed surface): y z D a The front face is at a distance of x/2 from P, and therefore: x D D + D S front front front front x D y z ,front x rate of change of with D x ,front 0 x x x 2 x D x + x D 0 x 2 President University Erwin Sitompul EEM 4/5
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element We have now, for front surface: x D + D x front y z x 0 x 2 In the same way, the integral over the back surface can be found as: ( ) x y z D a D y z D S back back back back ,back x D x x x D D ,back 0 x x 2 D x x back + y z x D 0 x 2 President University Erwin Sitompul EEM 4/6
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element If we combine the two integrals over the front and back surface, we have: x y z x D + x front back Repeating the same process to the remaining surfaces, we find: D y right left y + y x z D z bottom + z x y z top These results may be collected to yield: D D d x = D z D y + + x y z S x z y S D y D x D z D y + + S x d Q v z S President University Erwin Sitompul EEM 4/7
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element The previous equation is an approximation, which becomes better as v becomes smaller, and in the following section the volume v will be let to approach zero. For the moment, we have applied Gauss s law to the closed surface surrounding the volume element v. The result is the approximation stating that: D y D x D z y + + Charge enclosed in volume x v v z President University Erwin Sitompul EEM 4/8
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element Example Let D = y2z3ax+ 2xyz3ay+ 3xy2z2aznC/m2in free space. (a) Find the total electric flux passing through the surface x = 3, 0 y 2, 0 z 1 in a direction away from the origin. (b) Find |E| at P(3,2,1). (c) Find an approximate value for the total charge contained in an incremental sphere having a radius of 2 mm centered at P(3,2,1). D = S ( (a) d S S 1 ) ( 2 ) = + + 2 3 3 2 2 a a a a 2 3 y z xyz xy z dydz x y z x = = 0 0 z y = 3 x 1 2 = 2 3 0y z dydz 0 2 1 1 3 1 4 = 3 4 y z 0 0 2 3nC = President University Erwin Sitompul EEM 4/9
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element (b) + + 2 3 3 2 2 D = D = a 2 a a 2 a a 3 y z (2) (1) 4 = a xyz + xy z x y z + 3 3 2 2 a a 2(3)(2)(1) 36 z + a 3(3)(2) (1) P x y 2 z + 12 nC m x y D = + + 2 2 2 D = (4) (12) (36) P P = 2 38.158nC m =D P E P 38.158 nC m 8.854 10 4.31 kV m 0 2 = 12 = President University Erwin Sitompul EEM 4/10
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Application of Gauss s Law: Differential Volume Element ( D y (c) D x D z y + + x Q v z D y D x D z y + + x Q v z P P ) 3 3 4 3 0 2 + + (2 10 ) m 3 2 3 3 6 nC m xz xy z = = = 3 2 1 x y z ( ) 3 3 4 3 0 2(3)(1) + + (2 10 ) nC 3 2 6(3)(2) (1) 15 2.61 10 C President University Erwin Sitompul EEM 4/11
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Divergence We shall now obtain an exact relationship, by allowing the volume element v to shrink to zero. d D D D x y z v D S Q y + + = x z S v D S d D y D x D z Q y + + = = lim v lim v x z S v v 0 0 The last term is the volume charge density v, so that: D D D x y z D S d y + + = = lim v x z S v v 0 President University Erwin Sitompul EEM 4/12
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Divergence Let us no consider one information that can be obtained from the last equation: d D D D x y z v D S y + + = lim v x z S 0 This equation is valid not only for electric flux density D, but also to any vector field A to find the surface integral for a small closed surface. d A A A x y z v A S y + + = lim v x z S 0 President University Erwin Sitompul EEM 4/13
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Divergence This operation received a descriptive name, divergence. The divergence of A is defined as: A A A S d = = Divergence of div lim v S v 0 The divergence of the vector flux density A is the outflow of flux from a small closed surface per unit volume as the volume shrinks to zero. A positive divergence of a vector quantity indicates a source of that vector quantity at that point. Similarly, a negative divergence indicates a sink. President University Erwin Sitompul EEM 4/14
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Divergence D y D x D z Rectangular y = + + D div x z D 1 1 D z Cylindrical = + + D div ( ) D z D 1 r 1 1 Spherical = + + 2 D div ( ) (sin ) r D D r 2 sin sin r r r President University Erwin Sitompul EEM 4/15
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Divergence Example If D = e xsiny ax e xcosy ay+ 2z az, find div D at the origin and P(1,2,3). D y D x D z = 2 y = + + = + + x x D sin sin 2 e y e y div x z Regardless of location the divergence of D equals 2 C/m3. President University Erwin Sitompul EEM 4/16
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Maxwell s First Equation (Electrostatics) We may now rewrite the expressions developed until now: d = D S D div lim v S v 0 D y D x D z y = + + D div x z Maxwell s First Equation Point Form of Gauss s Law v = D div This first of Maxwell s four equations applies to electrostatics and steady magnetic field. Physically it states that the electric flux per unit volume leaving a vanishingly small volume unit is exactly equal to the volume charge density there. President University Erwin Sitompul EEM 4/17
Chapter 3 Electric Flux Density, Gauss s Law, and DIvergence Homework 4 D3.6. D3.7. D3.8. For D3.6., Replace P(2, 1,3) with P(StID, 1,StID+3). StID is the last two digits of your Student ID Number. Example: Someone with the Student ID of 002201700006 will do D3.6 with P(6, 1,9). All homework problems from Hayt and Buck, 7th Edition. Due: Monday, 27 April 2015. President University Erwin Sitompul EEM 4/18
