Physics
Charge Distribution
Charge distribution refers to the arrangement of electric charges within a given space. It describes how the charges are distributed, whether evenly or unevenly, and can be represented by charge density. Understanding charge distribution is crucial in analyzing the behavior of electric fields and the interactions between charged particles.
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7 Key excerpts on "Charge Distribution"
eBook - ePub- A. L. Stanford, J. M. Tanner(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
12Calculation of Electric Fields
Publisher Summary
This chapter explains several techniques appropriate to the calculation of electric fields. Electric fields play a key role in electromagnetic physics. It is, therefore, important to be able to determine the electric fields associated with a wide variety of Charge Distributions. Certain symmetrical arrangements of charge produce electric fields that may be determined by a simple, powerful, problem-solving procedure. This process is based on a relationship called Gauss’s law, which, in turn, is based on the concept of electric flux. Electric fields and the way that electric fields are related to Gauss’s law permit the understanding of how static charges position themselves in and on conducting materials. A continuous Charge Distribution may be considered a collection of infinitesimal elements of charge, each of which causes an electric field at a field point located a distance from the charge element. Each element of charge within the continuous distribution of charge is treated as if it were a point charge causing an electric field. The summation of infinitesimal elements is, of course, an integration process.Electric fields play a key role in electromagnetic physics. It is, therefore, important that we be able to determine the electric fields associated with a wide variety of Charge Distributions. This chapter introduces several techniques appropriate to the calculation of electric fields that result from relatively simple distributions of charge.Our first consideration is the calculation of electric fields at points in space caused by collections of fixed point charges. This procedure will then be extended to include the calculation of electric fields caused by continuous Charge Distributions.Certain symmetrical arrangements of charge produce electric fields that may be determined by a simple, powerful, problem-solving procedure. This process is based on a relationship called Gauss’s law, which, in turn, is based on the concept of electric flux. Therefore, we will illustrate electric flux and see how it is used in Gauss’s law to calculate the electric fields of symmetrical Charge Distributions.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
The total charge of an electrically closed system, or the algebraic sum of all its charges, remains constant. This statement is obvious if there are no elementary particle reactions. However, the law of conservation of charge is fundamental — it holds even if charged particles are created or destroyed. ◮ Charge Distribution. The spatial distribution of charge is characterized by its volume density ρ ( r ), measured in C / m 3 , surface density σ ( r ), measured in C / m 2 , and linear density λ ( r ), measured in C / m. We have dq = ρdV , dq = σdS , dq = λdl . (P3. 1 . 1 . 1 ) ◮ Dipole. Dipole moment. A system of charges is called electrically neutral if its net charge is zero. An electric dipole is a simple example of an electrically neutral system; it consists of two point charges, q and – q , separated by a distance l . The electric dipole moment is a vector quantity defined as p = q l , (P3. 1 . 1 . 2 ) where l = r + – r – is the displacement vector from the charge – q to the charge q . The dipole moment is measured in C m. 469 470 E LECTRODYNAMICS The following definition is a generalization to the case of an electrically neutral system: p = summationdisplay i q i r i = q ( r + – r – ), where q = ∑ q i > 0 q i is the sum of all positive charges, r + = ∑ q i > 0 q i r i /q is the center of positive charges, and r – = – ∑ q i < 0 q i r i /q is the center of negative charges. P3.1.2. Coulomb’s Law Coulomb’s law describes the electrostatic interaction of point charges , i.e., charged particles and bodies whose dimensions can be neglected as compared to the distances between them. Coulomb’s law states that, in vacuum, the electrostatic force, F , acting on a point charge q 1 due to the presence of a point charge q 2 is directly proportional to each of the charges and inversely proportional to the square of the distance r between them: F = k | q 1 | | q 2 | r 2 , (P3. 1 . 2 . 1 ) where k is the proportionality coefficient, dependent on the unit system chosen.
eBook - PDFZeta Potential in Colloid Science
Principles and Applications
- Robert J. Hunter, R. H. Ottewill, R. L. Rowell(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
Chapter 2 Charge and Potential Distribution at Interfaces 2.1 The electrostatic potential of a phase Before we can describe the potential distribution at an interface in a satis-factory way, we must examine some basic concepts in the theory of electro-statics. Our description will be based on the distinctions introduced by Lange, as outlined by Overbeek (1952, p. 124), and subsequently discussed in some detail in a number of review articles (see, Grahame, 1947, and, particularly, Parsons, 1954) and texts (e.g. Davies and Rideal, 1963, Sparnaay, 1972a). At the surface of any phase, even a pure metal in vacuo, there is a separation of positive and negative charge components so as to create a region of varying electrical potential which extends over a distance of the order of one or more molecular diameters. The potential differences generated across these layers are calculated to be of the order of a volt. Bardeen (1936), for example, calculated by wave mechanical methods that the electrons on a metal surface tend to protrude outwards from the surface, producing a negative charge layer which is compensated by a similar positive layer just inside the surface. (This conclusion had evidently been arrived at earlier by Frenkel (Frumkin and Pleskov, 1973)). By the same token, when two phases are in contact there is a tendency for any charged constituents, either electrons or ions, to be attracted, to different degrees, into the two phases and for surface dipolar molecules to be oriented selectively with respect to the two phases. The resulting electric field may also cause polarization effects in neighbouring molecules. All of these effects tend to produce a difference in the electrical potential between the interiors of the two phases. This difference is called the inner or Galvani potential difference, Αφ.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
2.11 Field and Potential of Continuous Charge Distributions The direct evaluation of the electrostatic field generated by continuous charge distribu- tion is practically feasible only if a symmetry of the system, allowing to simplify the calcu- lations, is identified. Then an evaluation can be done only for few cases and for some of their variants. a) Uniformly Charged Indefinite Straight Wire A uniformly charged indefinite wire is a very long, rectilinear and unidimensional, meaning that it is so thin that its transverse size is negligible, object carrying charge depos- ited according to a constant linear charge density λ. The electrostatic field generated by a wire in a point P is given by the sum of the contributions d E of all the infini- tesimal charge elements dq distributed along the wire, that is E P = d E B ∫ = 1 4πε 0 dq r 2 u B ∫ = 1 4πε 0 λdz r 2 u wire ∫ , where axis z has been chosen directed along the wire upwards and u is a unit vector, whose direction is variable and defined by the line connecting the position of element dq and point P, oriented positive away from the wire. Consider a cylindrical coordinate reference system featuring its axis z parallel to the wire and whose origin is chosen on the point of the wire where the normal to the wire pass- ing by P crosses it. + + + + + + + + + + λ P dq dz u dE Electrostatic Field and Electrostatic Potential Chapter 2 26 The indefinite length of the wire implies that each charge element dq1 = λ dz, depos- ited on an infinitesimal length dz of the wire in position +z, has a symmetrical element dq2 = dq1 in position –z. Hence, the electro- static field is given by the sum of the contri- butions of pairs of identical charge elements lying in symmetrical positions with respect to axis z. Each pair of charge elements generates an electrostatic field d E = d E 1 + d E 2 = dE u r , because the dEz components of the field gen- erated by each element are equal and opposite.
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
In order to find the electric field due to a continuous Charge Distribution, we have to define the following terms: 1. Linear charge density: If the charge is uniformly distributed along a line with charge per unit length λ ( r ) having unit of C . m − 1 , then dq = λdl , in which dl is an element of length along the line. The electric field of a line charge is E ( r ) = 1 4 π 0 P λ ( r ) r − r | r − r | 3 dl (2.20) 2. Surface charge density: If the charge is smeared out over a surface with charge per unit area σ ( r ) having unit of C . m − 2 , then dq = σd S , where d S is an element of surface along the area on the surface. The differential element of area on a spherical surface in spherical coordinates is d S = r 2 sin θdθdφ , in which ( θ, φ ) are the polar coordinates. The electric field for a surface charge is E ( r ) = 1 4 π 0 S σ ( r ) r − r | r − r | 3 d S (2.21) 3. Volume charge density: If the charge fills a volume with charge per unit volume ρ ( r ) having unit of C . m − 3 , then dq = ρd V , with d V as an element of volume. The electric field for a volume charge is E ( r ) = 1 4 π 0 V ρ ( r ) r − r | r − r | 3 d V (2.22) with V as the volume of the charged object. 1 Integration over a distribution of charge, dq , can be performed in the following ways: • for a line charge: dq = λdx , • for a sheet of charge: dq = σd S , and • for volume of charge: dq = ρd V , in which λ , σ , and ρ are the charges per unit length, per unit area, and per unit volume, respectively. Depending on the symmetry of the problem, we may choose to integrate over Cartesian coordinates or polar coordinates, for example, • for a rectangular sheet of charge: dq = σd S = σdxdy and • for a circular sheet of charge: dq = σd S = σrdrdθ . Electromagnetic Waves 81 Here, dq → λdl ∼ σd S ∼ ρd V 2.1.4 Electric Flux and Gauss’ Law The term flux is a quantity expressing the strength of a field of force in a given area.
No longer available |Learn more- James Babington(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
CHAPTER 7DISTRIBUTIONS OF CHARGE, MACROSCOPIC MATTER, AND BOUNDARY CONDITIONSIn this chapter, we will consider the implications of how real matter interacts with the physical electromagnetic field. There are many subtle problems associated in trying to understand the basic mechanisms. We want to be able to think about this without resorting to any sort of “fundamental” microscopic theory. The principle difficulty one encounters is in applying electromagnetism to account for structure of matter. The question is that if Coulomb law holds at the microscopic level then what is the underlying stability of matter - why doesn’t everything just collapse in on itself or blow apart. As the reader is no doubt aware, it is in the arena of quantum mechanics that this has to be sorted out. However, this book is on the classical theory of electromagnetic fields. One must be careful not to tread in other pastures, lest we not treat the topic consistently.Thinking about this then somewhat further, one can ask how the fields change when they enter into or are confined in normal solid matter. In such circumstances one needs to know what happens at the separating intermediate surface between two such media. If we were to regard Coulomb’s law as something very basic that would continue to be true at microscopic distances (so that it accounts for the structure of matter), then as we look at smaller and smaller length scales, how does the distribution of charge appear? What happens to this Charge Distribution as we change length scale? Necessarily, this will have to be an in part phenomenological approach because we do not know the microscopic theory. So the basic physical observables and measurements should not lose sight of the phenomenological assumptions.As in previous chapters, the setup and formalism is such that the electric and magnetic cases are treated in an analogous mirror like fashion. The astute reader has probably become aware that the magnetic cases tend to be a little bit more complicated than the equivalent electric one. The basic reason for this, in a general sense, is that for the magnetic field we are always finding a rotational angular momentum type variable that is intertwined with the magnetic field. In the electric case, it is typically a linear distance or velocity that has the equivalent connection.
- Nima Gharib, Javad Farrokhi Derakhshandeh, Peter Radziszewski(Authors)
- 2022(Publication Date)
- Elsevier(Publisher)
However, it makes no mention of the test charge Q. The electric field is a vector quantity that varies across points and is governed by the arrangement of source charges as shown in Fig. 3.1 ; physically, E (r) is the force per unit charge that would be exerted on a test charge if placed at P. 4. Continues Charge Distributions It is worth to emphasize that the electric field, e.g., Eq. (3.4), it is supposed that the field originates from a collection of discrete point charges denoted by q i. Thus, in terms of continues Charge Distributions, the electric field can be evaluated by the following integral: (3.5) Figure 3.1 Particle location in 3D cartesian system. Figure 3.2 Different types of charge. If the charge is spread along a line with charge/length (see Fig. 3.2B), then, the charge can be evaluated as dq = λdl′. Here, item dl ′ represents a length element along the line. If the charge is smeared across a surface (Fig. 3.2C), then, dq = σda′, where da is an area element on the surface; and if the charge fills a volume (Fig. 3.2D), then dq = ρ dτ ′, where d τ ′ is a volume element. Therefore, the electric field of a line, a surface, and a volume can be formulated as follows, respectively: (3.6) (3.7) (3.8) Eq. (3.8) is sometimes referred to as “Coulomb's law” due to its simplicity in comparison to the original and the fact that a volume charge is the most widespread and realistic situation. Please take notice of the definition of r in the following formulae. Initially, r i denoted the vector between the source charge q i and the field point r in Eq. (3.5). Similarly, in Eqs. (3.6)–(3.8), r denotes the vector connecting dq (and hence dl′, da′, or dτ′) to the field point r. 5. Field lines, flux, and Gauss's law In theory, we have concluded our discussion of electrostatics. Eq. (3.8) demonstrates how to calculate the field of a Charge Distribution, and Eq. (3.3) demonstrates how to calculate the force acting on a charge Q placed in this field
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