Electric Force

11 Key excerpts on "Electric Force"

• 

  eBook - ePub

  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  11

  Electric Charge and Electric Fields

  Publisher Summary

  This chapter begins our study of electric and magnetic phenomena, collectively called electromagnetism . Electromagnetic forces, like gravitational forces, are fundamental interactions that occur between particles. Just as mass is that property of matter that engenders gravitational attraction, electric charge is a property of certain fundamental particles, like electrons and protons, that causes electromagnetic interaction. Our investigation of electrostatics , the analysis of interactions between charges at rest, is based on a force law for charges and on the concept of electric fields. The force law for charges, called Coulomb’s law, is the electrical analogue of the law of universal gravitation. The electric field is an abstract concept used to characterize how the presence of charge alters the properties of the space around that charge.

  11.1 Electric Charge and Coulomb’s Law

  Electric charges exist in two forms, called positive charge and negative charge . The natural unit of charge is the positive charge of a proton or the negative charge of an electron. The SI unit of charge is the coulomb (C), which will be defined later in terms of electric current. This unusual definition is used because of the difficulty of making reproducible measurements on quantities of charge. For now, we will use as a measure of charge the experimentally obtained value for e , the magnitude of the charge of an electron or proton:

  e = 1.60 ×

  10

  − 19

  C

  (11-1)

  (11-1)

  Equivalently, a coulomb is the quantity of charge carried by 6.25 × 1018 electrons.

  An electron (or proton) bears the smallest unit of charge found in nature, so any quantity of charge q occurs as an integral number n of the basic charge e , or q = ne . Thus, as with any physical quantity made up of a number of “minimum parcels,” we say that electric charge is quantized; the least quantity that occurs, the quantum of charge, is the electronic charge e

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  eBook - ePub

  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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  Electric Field Analysis

  • Sivaji Chakravorti(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  It is the law that describes the electrostatic interaction between electrically charged particles. It was published by the French physicist Charles Augustin de Coulomb in 1785. He determined the magnitude of the Electric Force between two point charges using a torsion balance to study the attraction and repulsion forces of charged particles. The interaction between charged particles is a non-contact force that acts over some distance of separation. There are always two charges and a separation distance between them as the three critical variables that influence the strength of the electrostatic interaction. The unit of the electrostatic force, like all forces, is Newton. Being a force, the strength of the electrostatic interaction is a vector quantity that has both magnitude and direction.

  According to the statement of Coulomb’s law (1) the magnitude of the electrostatic force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of point charges and inversely proportional to the square of the separation distance between the point charges and (2) the electrostatic force of interaction acts along the straight line joining the two point charges. If the two point charges are of same polarity, the electrostatic force between them is repulsive; if they are of opposite polarity, the force between them is attractive.

  FIGURE 1.3 Electrostatic forces of interaction between two point charges.

  It should be noted here that two conditions are to be fulfilled for the validity of Coulomb’s law: (1) the charges involved must be point charges and (2) the charges should be stationary with respect to each other.

  Mathematically, the force between two point charges, as shown in Figure 1.3 , could be written as follows:

  F →

  2 1

  =

  ±

  Q 2

  ⋅

  Q 1

  4 π

  ε 0

  |

  r →

  2 1

  | 2

  u ˆ

  r 2 1

  and

  F →

  1 2

  =

  Q 1

  ⋅ ±

  Q 2

  4 π

  ε 0

  |

  r →

  1 2

  | 2

  u ˆ

  r 1 2

  sothat

  F →

  2 1

  = -

  F →

  1 2

  ⁢ ( 1.1 )

  where:

  • ε0 is permittivity of free space (≈8.854187 × 10–12

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  eBook - ePub

  Electromagnetism

  • I. S. Grant, W. R. Phillips(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  1 is

  In general, the force Fj on a charge

  qj

  due to a number of other charges

  qi

  is

  The symbol i ≠ j under the summation signs indicates that the summation for charge i is over all the other charges j , but of course not including charge i itself. This equation can be written in another way in terms of the position vectors of the charges with respect to a fixed origin O. If the position vectors of the charges q 1 , q 2 … q i … are r1 , r2 … ri … then the vector joining charges i and j is rij = rj – ri . The total force on

  qj

  is thus

  (1.3)

  A trivial example of the application of Equation (1.3) is in working out the electrostatic forces exerted by atomic nuclei containing many protons on the electrons surrounding them. Nuclei are much smaller than atoms, and for this purpose can be regarded as point charges. Equation (1.3) then tells us that the attractive force between an electron and a nucleus containing Z protons is Z times as great as that between an electron and a single proton.

  It turns out that apart from the sign, the charge carried by electrons and protons is the same, and has the magnitude e = 1.602 × 10−19 coulombs* : the charge on the proton is +e , that on the electron is – e . The strength of atomic interactions is governed by the size of the electronic charge e . Although e

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  Electrostatic Dust Mitigation and Manipulation Techniques for Planetary Dust

  • Nima Gharib, Javad Farrokhi Derakhshandeh, Peter Radziszewski(Authors)
  • 2022(Publication Date)
  • Elsevier

    (Publisher)

  Coulomb's law and electric field 3.1. Coulomb's law The basic electromagnetic concepts are studied one by one in sequence. A dimensional description will suffice in the vast majority of cases; the latter will be deduced from their relationship via the theory's basic equations, which will be tested experimentally. The next section will take place immediately after the current enumeration of essential concepts. To understand how much force does a single point charge q exert on a test charge Q while it is at rest r distance away, Coulomb's law can evaluate the answer as follows: (3.1) The constant ε 0 = (8.85 × 10 − 12 C 2 /N m 2) is referred to as open space permittivity (ludicrously). In SI units, newton (N) denotes force, meter (m) denotes distance, and coulomb denotes charge (C). In other words, the force is proportional to the charge product and inversely proportional to the square of the separation distance. Here, item Δ r is the separation vector with magnitude of r. The force is directed down the line q-Q ; it is repulsive if q and Q have the same sign, and attractive if they have the opposite sign. Coulomb's law and the principle of superposition provide the physical basis for electrostatics; all else is mathematical development of these basic laws, with the exception of a few specific features and elaboration these rules. 3.2. The electric field If there are many point charges, such as q 1, q 2, …, q n at distances r 1, r 2, …, r n from Q, the total force on Q is then equal to (3.2) or (3.3) Here, (3.4) Factor E (r) is referred to as the source of electric field charge, which is a function of position (r), since the separation vectors r i are location-dependent. 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

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  Electromagnetism

  Maxwell Equations, Wave Propagation and Emission

  • Tamer Becherrawy(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Chapter 2

  Electrostatics in Vacuum

  The interaction of electric charges, as expressed by Coulomb force, is formulated according to the Newtonian concept of action-at-a-distance: if a charge q′ is produced at r′ at a time t′, a charge q located at r feels the action of q′ instantaneously, whatever the distance |r − r′| and the medium that separates the charges. The concept of field was developed by Faraday, Maxwell, Lorentz, Einstein, and many others. In modern physics, all interactions are conceived as local, i.e. involving quantities defined at the same point r and at the same time t. Fields are physical entities that are endowed with energy, momentum, etc., and they may propagate with some finite speed as waves. Furthermore, in quantum theory, the same objects (electrons for instance) have both particle and wave properties.

  In this chapter, we introduce the concepts of electric field and potential, we derive the fundamental equations of electrostatics in vacuum, and we discuss some of their properties and the concept of electrostatic energy.

  2.1. Electric Forces and field

  In a famous experiment, Coulomb used a torsion balance to measure the force of interaction of electric charges. He verified that a small charge q1 acts on a small charge q2 situated at a distance r with a force FE = Ko q1 q2 /r2 oriented along the line joining the charges. This force is repulsive between like charges and attractive between unlike charges. It has a similar mathematical form to Newton's law of universal gravitation To specify both the direction and the magnitude, we write

  [2.1 ]

  Coulomb's force obeys the principle of action and reaction. Ko is a constant that depends on the adopted unit of charge. Using the coulomb (C) as the unit of charge and the Heaviside or rationalized system, we write

  [2.2]

  εo is the permittivity of vacuum. The factor 4π is introduced to simplify the writing of equations. The Electric Force is much more intense than the gravitational force and the coulomb is an enormous charge on the human scale: electric sparks are produced by less than one microcoulomb and rubbing produces a charge of the order of the nanocoulomb per square centimeter

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  Special Relativity

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  c . Why is this? A heavy emphasis on units and systems of measurement would be inappropriate in this chapter. The main interest lies elsewhere. On the other hand, the insight that relativity theory brings to electromagnetism removes much of the seeming arbitrariness of the subject. It exposes the essential relation between the Coulomb law and the magnetic force law, and between the constants that appear in them. And even at the outset it may be helpful to make some further remarks about these two basic force laws.

  When electrostatics was first developed, the Coulomb law, Eq. (8-1) , was used as a basis for defining the unit of electric charge. Since the CGS system for mechanical measurements was in general use at the time, the unit charge was defined as being of such a magnitude that, if placed 1 cm from a similar charge in vacuum, each charge would exert on the other a force of 1 dyne (= 10−5 newton). The force between two arbitrary charges in vacuum could then be simply set equal to q 1 q 2 /r 2 . Thus, using the CGS system, the unit of charge (1 esu) was defined in terms of the force between stationary charges.

  Much later it was recognized that, since currents are made up of moving charges, an alternative scheme for defining a unit of charge could be obtained from the analysis of the magnetic force exerted by one current on another. This was used to define the practical unit of charge (1 coulomb) that is the foundation of electrical measurements in the MKS system.1 In this scheme, with an appropriately defined unit of magnetic field, the basic law of force on a moving charge can then be written

  F mag

  =

  q 2

  u × B         ( MKS   system )

  ⁢

  ( 8-3 )

  1 The essential difference, of course, lies not in whether we use centimeters and grams rather than meters and kilograms, but in whether we base our definition of the unit charge on Coulomb’s law or on the force between currents.

  Given the value of B at a point, we can then calculate the magnetic force on any charged particle, with any velocity, at that point.

  The total electromagnetic force on a charge at a given point can then be written as the sum of the Electric Force that it would experience if stationary, plus the magnetic force as given by Eq. (8-3)

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  Static Fields and Potentials

  • Joy Manners(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)
• Charges exert a force on each other. The properties of charge are summarized in Box 1.1 of Section 3.2.
• Coulomb's law of force between charges situated in a vacuum may be summarized in the vector equation
  F el

  =

  1

  4 π

  ε 0

  (

  q 1

  q 2

  r 2

  )

  r ^

  .

  (1.13)
  The constant ɛ0 is called the permittivity of free space.
• The electric field
  ℰ

  ( r )
  (r) is a vector quantity, defined at all points in space. At any particular point r its value is given by the electrostatic force per unit charge at that point. So
  ℰ

  ( r )

  =

  F el

  (

  on   q   at   r

  )

  q

  .

  (1.14)
• The electrostatic force experienced by a point charge q at r is given by
  F el

  (

  on   q   at   r

  )

  = q ℰ

  ( r )

  .

  (1.15)
• The electric field due to a point charge Q at the origin is given by
  ℰ

  ( r )

  =

  (

  Q

  4 π

  ε 0

  r 2

  )

  r ^

  .

  (1.16)
• Fields of a given type, due to different sources, may be added vectorially using the triangle or parallelogram rule.
• The similarities between the interaction of masses via gravitational fields and the interaction of charges via electric fields are summarized in Box 1.2 below.
• Gravitational and electric fields can be represented pictorially by means of field lines.
• Applications and occurrences of electric fields are widespread. Discussed in this chapter are Millikan's oil drop experiment, ink-jet and electrostatic printing, and liquid crystal displays. The Earth has a naturally occuring electric field.