Physics
Divergence of Electrostatic Field
The divergence of an electrostatic field is a measure of how much the field spreads out from a point. It is a vector operation that describes the flux of the electric field through an infinitesimal surface surrounding a point. A positive divergence indicates that the field is spreading out, while a negative divergence indicates that the field is converging towards the point.
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5 Key excerpts on "Divergence of Electrostatic Field"
- K Umashankar(Author)
- 1989(Publication Date)
- WSPC(Publisher)
88 INTRODUCTION TO ENGINEERING ELECTROMAGNETIC FIELDS D«di cube S r ^ D x d°v 3D Z Q lim u u u c ^ = T ^ + T ^ + = r ^ l = l«rn [-7H (3.2.10b) Av-0 Av L 3x dy dz J Av-0 AV J The right hand side term in the above expression (3.2.10b) is the volume charge density at the center point P of the cubical volume. The left hand side term is the basic mathematical definition of the divergence of a vector quantity. The middle term is the divergence of the vector quantity D expressed in the rectangular coordinate system. As per discussion in the chapter 1, expression (1.6.3), let us denote the vector differential operator or del operator as * o *> o * o V = K r -+ U — + Z — (3.2.10c) dx dy dz Thus, we obtain the Gauss's law in the point or differential form, which states that the divergence of the electric flux density at any point in a three dimensional medium is equal to volume charge density at that point V*D(x,y,z) = p v (x,y,z) (3.2.11) Suppose at a given point in space the charge density is zero. Then the divergence of the electric flux density at that specific point is zero. Thus, the expression (3.2.11) enables one to test whether there exists a source or not at a given point in space. The divergence is always taken as positive for the case of positive charge density located at a point with electric field lines coming away from the point. If the electric field lines are converging or terminating at a point, then the divergence is taken as negative with negative charge density located at the point. Eventhough the expression (3.2.11) is obtained using rectangular coordinates, in fact, it is also valid in terms of cylindrical and spherical coordinate systems. The relevant expressions are stated in chapter 1, specifically the expressions (1.9.2a, b, c) give the divergence of a vector in other coordinate systems. Example 3.2.1 The electric field distribution in a three
eBook - PDFCalculus
Multivariable
- William G. McCallum, Deborah Hughes-Hallett, Daniel E. Flath, Andrew M. Gleason, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Ayse Arzu Sahin, Adam H. Spiegler, Jeff Tecosky-Feldman, Thomas W. Tucker, Aaron(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density. Figure 19.29: Vector field showing a source Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field , written div , is a scalar-valued function defined by div (, , ) = lim Volume→0 ∫ ⋅ Volume of . Here is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE DIVERGENCE OF A VECTOR FIELD 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If = 1 + 2 + 3 , then div = 1 + 2 + 3 . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div , namely ∇ ⋅ . Using ∇ = + + , we can write div = ∇⋅ = ( + + ) ⋅ ( 1 + 2 + 3 ) = 1 + 2 + 3 . Example 1 Calculate the divergence of ( ) = at the origin (a) Using the geometric definition.
eBook - PDFCalculus
Single and Multivariable
- Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum, Daniel E. Flath, Patti Frazer Lock, David O. Lomen, David Lovelock, Brad G. Osgood, Douglas Quinney, Karen R. Rhea, Jeff Tecosky-Feldman, Thomas W. Tucker, Otto K. Bretscher, Sheldon P. Gordon, Andrew Pasquale, Joseph Thrash(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density. Figure 19.29: Vector field showing a source Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field , written div , is a scalar-valued function defined by div (, , ) = lim Volume→0 ∫ ⋅ Volume of . Here is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE DIVERGENCE OF A VECTOR FIELD 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If = 1 + 2 + 3 , then div = 1 + 2 + 3 . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div , namely ∇ ⋅ . Using ∇ = + + , we can write div = ∇⋅ = ( + + ) ⋅ ( 1 + 2 + 3 ) = 1 + 2 + 3 . Example 1 Calculate the divergence of ( ) = at the origin (a) Using the geometric definition.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
These characteristics of the electrostatic field lead to an important mathematical relationship known as Gauss’s law. This law is named in honor of the extraordinary German mathematician and scientist Karl Friedrich Gauss (Figure 6.2). Gauss’s law gives us an elegantly simple way of finding the electric field, and, as you will see, it can be much easier to use than the integration method described in the previous chapter. However, there is a catch—Gauss’s law has a limitation in that, while always true, it can be readily applied only for charge distributions with certain symmetries. Chapter 6 | Gauss's Law 235 Figure 6.2 Karl Friedrich Gauss (1777–1855) was a legendary mathematician of the nineteenth century. Although his major contributions were to the field of mathematics, he also did important work in physics and astronomy. 6.1 | Electric Flux Learning Objectives By the end of this section, you will be able to: • Define the concept of flux • Describe electric flux • Calculate electric flux for a given situation The concept of flux describes how much of something goes through a given area. More formally, it is the dot product of a vector field (in this chapter, the electric field) with an area. You may conceptualize the flux of an electric field as a measure of the number of electric field lines passing through an area (Figure 6.3). The larger the area, the more field lines go through it and, hence, the greater the flux; similarly, the stronger the electric field is (represented by a greater density of lines), the greater the flux. On the other hand, if the area rotated so that the plane is aligned with the field lines, none will pass through and there will be no flux. Figure 6.3 The flux of an electric field through the shaded area captures information about the “number” of electric field lines passing through the area.
- S. B. Lal Seksena, Kaustuv Dasgupta(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Q = ∫∫∫ s ρ dv .......... 3.51 Substituting this in equation 3.50 we get, ∫∫ s E – . ⏐ ds — ⏐ .n ˆ = ∫∫ v ∫ ρ dv .......... 3.52 Applying divergence theorem on left handed side ∫∫ v ∫( ∇ – .E – ) dv = ∫∫ v ∫ ρ dv .......... 3.53 Hence, ∫∫ v ∫ ( ∇ – .E – – ρ ) dv = ∫∫ v ∫ dv .......... 3.54 Since this is true for any arbitrary volume, the integral must be vanished identically Hence, ∇ -= ∇ = . . E E o o ρ ε ρ ε 0 .......... 3.55 .......... 3.56 Equation represents the Gauss’s law in three dimensional free space 3.7 Poisson’s Equation in Electrostatic Field Equation 3.56 establishes the relation between electrical field intensity and electric potential in three dimension. E – = – ∇ – V If we substitute this in equation for electric field intensity we have ∇ .(– ∇ V) = ρ / ε 0 .......... 3.57 Fig. 3.6: Gauss’s law for volume charge dv v S 140 Fundamentals of Electrical Engineering or, -∇ = 2 V o ρ ε .......... 3.58 or, ∇ = -2 V o ρ ε .......... 3.59 Equation 3.59 is a typical second order differential equation in three dimensional vector space. This equation is known as Poission’s equation, named after French mathematician Simeon Denis Poisson. 3.8 Capacitor and Capacitance The most useful application of electrostatic field for electrical engineering is capacitor. Two conductors separated by a dielectric medium form a capacitor. Capacitors have the capacity to store the static electric charges without leaking that through the dielectric medium Fig. 3.7 illustrates how a capacitor works. Let us suppose two conducting plates A and B are separated by a dielectric medium. If we give a charge +Q to the A plate both –Q and +Q charges will be inducted in plate B on the nearer and far side of the plate B, respectively. A plate B is earthed +Q will be repelled back to earth resulting plate B to be charged with –Q. If the electric potential of plate A due to charge +Q is +V, the electric potential of plate B due to charge –Q ought to be –V.
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