8 Key excerpts on "Electric Field of Multiple Point Charges"

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  eBook - ePub

  Electrical Engineering

  Fundamentals

  • Viktor Hacker, Christof Sumereder(Authors)
  • 2020(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  electrostatic field.

  Each electric charge is surrounded by an electric field. Electric charges interact and affect each other:

  • Like charges repel each other.
  • Unlike charges attract each other.

  A charged particle moves along an imaginary line in the electric field. Such lines of occurring forces are called electric field lines. The following applies for electric field lines:

  • Direction: The electric field lines start at the positive charge (source) and end at the negative charge (sink).
  • Electric field lines always start and end on the surface of an electric charge.
  • Electric field lines always leave the surface of the electric charge in a right angle.
  • Field lines do not intersect.
  • The denser the field lines, the stronger the force exerted on the charges.

  From a mathematical point of view, the electric field is a vector field of the electric field strength. It assigns a vector for direction and magnitude of electric field strength to each point in the space.

  Figure 1.14: Attracting force and Repulsive force of charges.

  The electric field is the cause for forces exerted on charges. According to the first of Newton’s axioms, a body accelerates as long as force is exerted on it. Moving charge carriers are called current. The sources of an electric field are positive charges, sinks and negative charges. The electric field [Vm−1 ; NC−1 ] is uniform when neither its magnitude nor its direction changes from one point to another.

  1.7.1  Force on charged particle in electric field

  The electric field strength is the measure of the force, a charged body experiences in an electric field. The existing electric field strength at any point in the field can be defined as the force exerted on a positive point unit charge

  Q +

  = 1   A s

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  Electromagnetics Explained

  A Handbook for Wireless/ RF, EMC, and High-Speed Electronics

  • Ron Schmitt(Author)
  • 2002(Publication Date)
  • Newnes

    (Publisher)

  By convention, the electric field is always drawn from positive to negative. It follows that the force lines emanate from a positive charge and converge to a negative charge. Furthermore, the electric field is a normalized force, a force per charge. The normalization allows the field values to be specified independent of a second charge. In other words, the value of an electric field at any point in space specifies the force that would be felt if a unit of charge were to be placed there. (A unit charge has a value of 1 in the chosen system of units.)

  Electric field = Force field as “felt” by a unit charge

  To calculate the force felt by a charge with value, q, we just multiply the electric field by the charge,

  The magnitude of the electric field decreases as you move away from a charge, and increases as you get closer. To be specific, the magnitude of the electric field (and magnitude of the force) is proportional to the inverse of the distance squared. The electric field drops off rather quickly as the distance is increased. Mathematically this relation is expressed as

  where r is the distance from the source and q is the value of the source charge. Putting our two equations together gives us Coulomb’s law,

  where

  q1

  and

  q2

  are the charge values and r is the distance that separates them. Electric fields are only one example of fields.

  OTHER TYPES OF FIELDS

  Gravity is another field. The gravitational force is proportional to the product of the masses of the two objects involved and is always attractive. (There is no such thing as negative mass.) The gravitational field is much weaker than the electric field, so the gravitational force is only felt when the mass of one or both of the objects is very large. Therefore, our attraction to the earth is big, while our attraction to other objects like furniture is exceedingly small.

  Another example of a field is the stress field that occurs when elastic objects are stretched or compressed. For an example, refer to Figure 2.2 . Two balls are connected by a spring. When the spring is stretched, it will exert an attractive force on the balls and try to pull them together. When the spring is compressed, it will exert a repulsive force on the balls and try to push them apart. Now imagine that you stretch the spring and then quickly release the two balls. An oscillating motion occurs. The balls move close together, then far apart and continue back and forth. The motion does not continue forever though, because of friction. Through each cycle of oscillation, the balls lose some energy until they eventually stop moving completely. The causes of fiction are the air surrounding the balls and the internal friction of the spring. The energy lost to friction becomes heat in the air and spring. Before Einstein and his theory of relativity, most scientists thought that the electric field operated in a similar manner. During the 1800s, scientists postulated that there was a substance, called aether, which filled all of space. This aether served the purpose of the spring in the previous example. Electric fields were thought to be stresses in the aether. This theory seemed reasonable because it predicted the propagation of electromagnetic waves. The waves were just stress waves in the aether, similar to mechanical waves in springs. But Einstein showed that there was no aether. Empty space is just that—empty.*

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  Basics of Electromagnetics and Transmission Lines

  • G. Jagadeeswar Reddy, T. Jayachandra Prasad(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 1 Static Electric Fields 1.1    Introduction Electrostatic in the sense static or rest or time in-varying electric fields. Electrostatic field can be obtained by the distribution of static charges. The two fundamental laws which describe electrostatic fields are Coulomb’s law and Gauss’s law: They are independent laws. i.e., one law does not depend on the other law. Coulomb’s law can be used to find electric field when the charge distribution is of any type, but it is easy to use Gauss’s law to find electric field when the charge distribution is symmetrical. 1.2    Coulomb’s Law This law is formulated in the year 1785 by Coulomb. It deals with the force a point charge exerts on another point charge; generally a charge can be expressed in terms of coulombs. 1 coulomb = 6 × 10 18 electrons 1 electron charge = − 1.6 × 10 − 19 Coulombs Coulomb’s law states that the force between two point charges Q 1 and Q 2 is along the line joining between them, directly proportional to the product of two point charges, and inversely proportional to the square of the distance between them ∴ F = K Q 1 Q 2 R 2 where K is proportional constant In SI, a unit for Q 1 and Q 2 is coulombs(C), for R meters(m) and for F. newtons(N). K = 1 4 π ∊ 0 where ∊ 0 = permittivity of free space (or) vacuum = 8.854 × 10 − 12 farads / meter = 10 − 9 36 π farads / m K = 36 π 4 π × 10 − 9 = 9 × 10 9 m / farads F = Q 1 Q 2 4 π ∊ 0 R 2 (1.2.1) Assume that the point charges Q 1 and Q 2 are located at (x 1, y 1,. z 1) and (x 2, y 2, z 2) with the position vectors r ¯ 1 and r ¯ 2 respectively. Let the force on Q 2 due to Q 1 be F ¯ 12 which can be written as F ¯ 12 = Q 1 Q 2 4 π ∊ 0 R 2 a ¯ R 12 (1.2.2) where a ¯ R 12 is unit vector along the vector R ¯ 12. Graphical representation of the vectors in rectangular coordinate system is shown in Fig.1.1 Fig

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  Pocket Book of Electrical Engineering Formulas

  • Richard C. Dorf, Ronald J. Tallarida(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  26 Static Electric Fields 1.  Unit Vectors and Coordinate Systems The unit vectors for the Cartesian (rectangular) system shown in Figure 26.1(a) are a x, a y, a z and all three vectors are constant. The unit vectors for the cylindrical coordinate system shown in Figure 26.1(b) are a ρ, a ϕ, a z where a z is constant. The unit vectors for the spherical coordinate system shown in Figure 26.1(c) are a r, a θ, a ϕ 2.  Coulomb’s Law For two-point charges Q 1 the source of the field force F, and Q we have the force. on Q as F = Q 1 Q 4 π ∈ 0 R 2 a R (N) where a R is the vector of unit length pointing from Q 1 to Q, ϵ 0 = 10 −9 (36 π), and R is the distance between the charges. FIGURE 26.1. Unit vectors for (a) Cartesian, (b) cylindrical, and (c) spherical coordinates. Electric Field Intensity The electrostatic field intensity is defined as the force on Q when Q = 1 C so E = Q 1 4 π ϵ 0 R 2 a R (V / m) and F = Q E (N) 3.  Gauss’ Law Electric flux. density, D, is D = ϵ 0 E (C / m 2) Gauss’ law states that the net flux of D, or electric flux ψ passing through a surface is equal to the net positive charge enclosed within the surface and thus ψ = ∮ D s ⋅ d S = Q where D s, is the value of D at the surface and dS is the surface element. 4.  Maxwell’s Equation (Electrostatics) The electric flux per unit volume leaving a vanishingly small volume unit is equal to the volume charge density there: div D = ρ where div is divergence and ρ is a volume charge density. Using div D = ∇ · D, we have ∇ ⋅ D = ρ 5.  Poisson’s Equation ∇ · ∇ V = − ρ ∈ or ∇ 2 V = − ρ ε where E = − ∇ V. 6.  Current Density The current density J is related to the electric field E for a metallic conductor as J = σ E where σ is the conductivity of the conductor. The current density J is a convection current J = ρ υ where υ is a velocity vector and ρ is the volume charge density.

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  Radiation Detection

  Concepts, Methods, and Devices

  • Douglas McGregor, J. Kenneth Shultis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 8 Essential Electrostatics

  I happen to have discovered a direct relation between magnetism and light, also electricity and light, and the field it opens is so large and I think rich.

  Michael Faraday

  Numerous radiation detectors rely on the current produced by the motion of mobile charges created by radiation events. Radiation interactions within the detector volume ionize the absorbing medium, thereby creating mobile point charges, usually referred to as charge carriers (because these particles carry the charge through the detector). The detector may be a solid, liquid or gas, within which two or more electrodes are installed. Generally, a voltage is applied to the detector electrodes to move, or drift, the mobile charges across the detector volume to produce the current.

  Electronic detectors range from simple radiation counters to complex radiation spectrometers. The former are used to indicate the presence of ionizing radiation, whereas the latter can also identify the energies and types of radiation. In either case, the fundamental physics of current induction formed by mobile charge carriers applies. Introduced in this chapter are the basic concepts governing charge and current induction in electronic radiation detectors.

  8.1Electric Field

  Consider a point charge of Q coulombs, as shown in Fig. 8.1 . The charge itself has electric field lines emanating from it radially outward (or inward). If a point charge is centered within an imaginary spherical surface, the electric field lines intersect the surface at right angles (perpendicular), with the imaginary surface area 4πr2 , where r is the radius of the sphere. Suppose the number of field lines is denoted by N, then the surface density of field lines becomes N/(4πr2 ) per unit area. It is easy to understand that the electric field line density decreases with r2

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  Transmission Lines and Wave Propagation

  • Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  It is helpful to review some of the basic concepts of the electromagnetic fields, with emphasis on those aspects which bear directly on wave propagation. In so doing it will be assumed that the reader has studied (a) the operations and functions of vector analysis (key definitions and formulas are summarized in Appendix A) and (b) the basic concepts of static and quasistatic electric and magnetic field theory. The latter will be recapitulated here in order to provide a logical base for applications of Maxwell's equations and the accompanying boundary conditions.

  The following elementary phenomena indicate the space pervasiveness of electromagnetic effects:

  1. The presence of current in two circuits in proximity to each other is accompanied by mechanical forces on each conductor, forces which change if either current is changed.
  2. A changing of the current in either of two such circuits is accompanied by an induced voltage in the other.
  3. Capacitors consisting of metallic spheres or other conducting bodies suspended in vacuum or in an insulating medium may be charged and later discharged. During these processes wire-borne current flows onto one sphere and off the other.
  4. The presence of electric charges on two bodies is accompanied by a mechanical force on each, forces which change if either charge is changed.

  Mechanical-force effects, items 1 and 4, have been mentioned primarily because they help in assigning directions to the fields. Item 2 states an application of Faraday's law, and item 3 describes a situation in which the displacement current, postulated by Maxwell, complements a discontinuous conduction current to yield, as a resultant, a composite current which is continuous.

  a. Directional Properties of the Electric and Magnetic Fields

  The direction to be assigned to a vector field is an important part of the definition of the function and is chosen in the light of the physical situation to which it relates. It is rather obvious for the electric field, but less so for the magnetic field.

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  Fields of Force

  The Development of a World View from Faraday to Einstein.

  • William Berkson(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Maxwell assumed that the total displacement (D) is directly proportional to the force acting on the ball; the constant of proportionality is analogous to the dielectric constant or specific inductive capacity (ε) of the medium (D =ε E). The energy of the electrical field will be analogous to the elastic energy of the distorted particles. This energy must be equal to the work done in distorting the particles—the force exerted by the vortices multiplied by the distance the material is displaced (which is proportional to ED =ε E 2). In Maxwell’s model charge is supposed to be caused by a pressure exerted by the electrical particles upon each other. The pressure is analogous to electrical potential or tension (ψ). The difference in the pressure on two sides of an electrical partiele will give rise to the part of electromotive force caused by static electricity. A charged body is one whose electrical particles exert a net pressure upon the particles of the surrounding dielectric. The cause of this pressure, we should note, must be alien to the mechanism itself. Maxwell derived his equations in three stages. First, he used the assumption of the vortices to account for purely magnetic effects. Second, he used the assumption of the electrical balls to derive the relations between current and magnetism, including induction. Third, he used the assumption of elasticity of the electrical balls to account for the effects of static charge. Each of these stages were steps towards Maxwell’s crowning achievement: the electromagnetic theory of light. The idea of the initial stage is very ingenious; the development of the idea is relatively straightforward. As mentioned before, Maxwell took as his starting point Faraday’s suggestion that there is a tension along the lines of force and a pressure between them

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  Polarized Light

  • Dennis H. Goldstein(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  The reason for treating the motion of a charge in this chapter as well as in the previous chapter is that the material prepares us to understand and describe the Lorentz–Zeeman effect. Another reason for discussing the motion of charged particles in the electromagnetic field is that it has many important applications. Many physical devices of importance to science, technology, and medicine are based on our understanding of the fundamental motion of charged particles. In particle physics these include the cyclotron, betatron (invented by Don Kerst, one of the author’s professors at the University of Wisconsin–Madison), and synchrotron, and in microwave physics the magnetron and traveling-wave tubes. While these devices will not be discussed here, the mathematical analysis presented is the basis for describing all of them. Our primary interest is to describe the motion of charges as they apply to atomic and molecular systems and to determine the intensity and polarization of the emitted radiation.

  In this chapter, we treat the motion of a charged particle in three specific configurations of the electromagnetic field: (1) the acceleration of a charge in an electric field; (2) the acceleration of a charge in a magnetic field; and (3) the acceleration of a charge in perpendicular electric and magnetic fields. In particular, the motion of a charged particle in perpendicular electric and magnetic fields is extremely interesting not only from the standpoint of its practical importance but because the paths taken by the charged particle are quite beautiful and remarkable. Much of this material is taught in courses on plasma physics, since trajectories and containment of particles are so important to that field.

  In an electromagnetic field, the motion of a charged particle is governed by the Lorentz force equation:

  F = q

  [

  E +

  (

  v × B

  )

  ]

  ,

  ⁢ ( 30.1 )

  where q is the magnitude of the charge, E is the applied electric field, B is the applied magnetic field, and v is the velocity of the charge. The background to the Lorentz force equation and the phenomenon of the radiation of accelerating charges can be found in the texts given in the references [1-8]. The text by G. P. Harnwell [3 ] on electricity and magnetism is especially clear and illuminating.

  30.1.1 Motion of an Electron in a Constant Electric Field

  The first and simplest example of the motion of an electron in an electromagnetic field is for a charge moving in a constant electric field. The field is directed along the z axis and is of strength E0 . The vector representation for the general electric field E

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