Work in Electrostatics

8 Key excerpts on "Work in Electrostatics"

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  Introduction to Electrodynamics

  • David J. Griffiths(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  2.4 WORK AND ENERGY IN ELECTROSTATICS 2.4.1 The Work It Takes to Move a Charge Suppose you have a stationary configuration of source charges, and you want to move a test charge Q from point a to point b (Fig. 2.39). Question: How much work will you have to do? At any point along the path, the electric force on Q is F = QE; the force you must exert, in opposition to this electrical force, is −QE. (If the sign bothers you, think about lifting a brick: gravity exerts a force mg downward, but you exert a force mg upward. Of course, you could apply an even greater force—then the brick would accelerate, and part of your effort would be “wasted” generating kinetic energy. What we’re interested in here is the minimum force you must exert to do the job.) The work you do is therefore W =  b a F · d l = −Q  b a E · d l = Q[V (b) − V (a)]. Notice that the answer is independent of the path you take from a to b; in mechan- ics, then, we would call the electrostatic force “conservative.” Dividing through by Q, we have V (b) − V (a) = W Q . (2.38) In words, the potential difference between points a and b is equal to the work per unit charge required to carry a particle from a to b. In particular, if you want to bring Q in from far away and stick it at point r, the work you must do is W = Q[V (r) − V (∞)], q 1 q 2 q i a b Q FIGURE 2.39 92 Chapter 2 Electrostatics so, if you have set the reference point at infinity, W = QV (r). (2.39) In this sense, potential is potential energy (the work it takes to create the system) per unit charge (just as the field is the force per unit charge). 2.4.2 The Energy of a Point Charge Distribution How much work would it take to assemble an entire collection of point charges? Imagine bringing in the charges, one by one, from far away (Fig. 2.40). The first charge, q 1 , takes no work, since there is no field yet to fight against.

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  Steady Electric Fields and Currents

  Elementary Electromagnetic Theory

  • B. H. Chirgwin, C. Plumpton, C. W. Kilmister(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The action on any finite distribution of charge could be obtained by combining these forces using integrations where necessary to give a total resultant force at any point together with a couple. Our aim in this discussion of energy and its relation to the forces which act upon bodies and charges may be roughly stated as follows. The charges which are the sources of an electrostatic field are regarded as having been assembled in their final positions from dispersal at infinity. In assembling the charges work was done against the mutual interactions of the charges, and this work is regarded as the potential energy of the field. If the configura-tion of the field alters, the forces acting on any body, or charge, in the field do work, and this work is done at the expense of the potential energy. The con-figuration of any field, i.e. the positions and orientations of conductors, the positions of point charges, etc., maybe specified by a number of generalized coordinates (see CM., vol. 3, chap. 2 ; vol. 6, chap. 2) of which we take x to be typical. Corresponding to each of these coordinates is a generalized force X which tends to increase x. (The external agents holding the whole system stationary have to exert a force —Xto balance each force A exerted by the field.) The forces exerted by the field are often called ponderomotive forces. In a change of configuration in which each x increases by δχ the pondero-motive forces do work ^Χδχ; this work is done at the expense of the poten-tial energy W. Therefore, -bW = Σ χδχ · ( 3 · 16 ) When the different coordinates x are independent we deduce that We consider now a number of important special cases. (3.17) 92 ELEMENTARY ELECTROMAGNETIC THEORY 1. A POINT CHARGE IN A FIELD The work necessary to place a charge Q at a point P is W=QV, (3.18) where Fis the potential of the field at P.

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  University Physics Volume 2

  • William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
  • 2016(Publication Date)
  • Openstax

    (Publisher)

  But energy and voltage are not the same thing. A motorcycle battery, for example, is small and would not be very successful in replacing a much larger car battery, yet each has the same voltage. In this chapter, we examine the relationship between voltage and electrical energy, and begin to explore some of the many applications of electricity. Chapter 7 | Electric Potential 285 7.1 | Electric Potential Energy Learning Objectives By the end of this section, you will be able to: • Define the work done by an electric force • Define electric potential energy • Apply work and potential energy in systems with electric charges When a free positive charge q is accelerated by an electric field, it is given kinetic energy (Figure 7.2). The process is analogous to an object being accelerated by a gravitational field, as if the charge were going down an electrical hill where its electric potential energy is converted into kinetic energy, although of course the sources of the forces are very different. Let us explore the work done on a charge q by the electric field in this process, so that we may develop a definition of electric potential energy. Figure 7.2 A charge accelerated by an electric field is analogous to a mass going down a hill. In both cases, potential energy decreases as kinetic energy increases, – ΔU = ΔK . Work is done by a force, but since this force is conservative, we can write W = – ΔU . The electrostatic or Coulomb force is conservative, which means that the work done on q is independent of the path taken, as we will demonstrate later. This is exactly analogous to the gravitational force. When a force is conservative, it is possible to define a potential energy associated with the force. It is usually easier to work with the potential energy (because it depends only on position) than to calculate the work directly.

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  Electromagnetic Field Theory Fundamentals

  • Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  In order to avoid consideration of any kinetic energy that may be acquired by a moving charge, we shall assume that the external force just balances the electric force, as depicted in Figure 3.19b. In that case, dW = − q E · d Figure 3.19 Motion of a test charge in an electric field caused by (a) the electric field and (b) the external force The total work done by the external force in moving the test charge from point b to a is W ab = − q a b E · d (3.30) If we move the charge around a closed path, as indicated in Figure 3.20, the work done must be zero. In other words, c E · d = 0 (3.31) which simply states that the E field under static conditions is irrotational or conservative. However, a field is conservative if its curl is zero. Thus, ∇ × E = 0 (3.32) If the curl of a vector field is zero, the vector field can be represented in terms of the gradient of a scalar field. Thus, we can express the E field in terms of a scalar field V as E = −∇ V (3.33) The reason for the minus sign will soon become evident. Figure 3.20 Movement of charge q along a closed path c in an electric field 87 3.5 The electric potential We can now express (3.30) as W ab = − q a b E · d = q a b ∇ V · d Substituting ∇ V · d = dV (Section 2.8), we have W ab = − q a b E · d = q V a V b dV = q [ V a − V b ] = qV ab (3.34) where V a and V b are the values of the scalar field V at points a and b . We speak of V a and V b as the electric potentials at points a and b , re-spectively, with respect to some reference point. It is clear that V ab = V a − V b defines the potential of point a with respect to point b (this is called the potential difference between the two points). If the work done is positive, then the potential at point a is higher than that at point b . In other words, when the external force is pushing the positive charge against the E field, the potential energy of the charge is increasing. That is why we have used the negative sign in (3.33).

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  Electromagnetism

  Maxwell Equations, Wave Propagation and Emission

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

    (Publisher)

  binding energy of the system of bound charges.

  In the case of a continuous charge distribution with a density qv (r) (Figure 2.11b ), an infinitesimal volume near ri contains the charge . The interaction energy of and is and the total interaction energy is

  [2.66 ]

  In the second expression, is the potential produced at r by all the charge of . As in [2.65 ], in principle, we should consider different elements of volume d i and d j and use the potential V′(r) of the whole volume except d . However, the potential of d tends to zero with the dimensions of d (see, for instance, equation [2.59 ] in the case of a small sphere of radius R). Thus, we may use the total potential V instead of V′. Similar relations to [2.66 ] may be written in the case of a surface charge but not in the case of a linear charge as the potential of an element of length dL does not go to zero with dL (see section 2.7 and Problem 2.18 ).

  Using Gauss's law, we may write the electrostatic energy [2.66 ] in a form that uses only the potential and the field:

  [2.67 ]

  where we have used the relation

  If we are only interested in the total electrostatic energy of a charge distribution that occupies a finite , the first term of equation [2.67 ] is the integral of the divergence of (VE) over this volume. It may be transformed into the outgoing flux of (VE) from a large surface S enclosing the system on which V and E go to zero rapidly enough for the flux to be zero1

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  General Physics Electromagnetism Optics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Its definition is given by the relationship  F = −  ∇U e . Since a coulomb force has only its radial component, this relationship is expressed just as F r = − dU e dr ⇒ dU e = − F r dr that is, by choosing as U e ∞ ( ) = 0 the arbitrary constant which characterises any potential energy, the associated potential is given by U e = − F r dr ∞ r ∫ = − 1 4πε 0 q ′ q r 2 dr ∞ r ∫ = 1 4πε 0 q ′ q r . A point-like charge q generates an electrostatic field defined by  E =  F ′ q = −  ∇ U e ′ q ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , so,by defining the electrostatic potential † of a point-like charge as V = U e ′ q = 1 4πε 0 q r , the relationship  E = −  ∇V between an electrostatic field and its associated electrostatic potential is obtained. In any field of conservative forces, work and potential energy are related by the rela- tionship W = − Δ U. Then, the work done by an electrostatic force  F = ′ q  E for an infini- tesimal displacement from point P to point ′ P of a charge ′ q can be written as Chapter 2 Electrostatic Field and Electrostatic Potential 19 † Potential and volume share the same symbol. In order to mark a difference, this textbook uses character V for potential and character V for volume. The characters are similar, so the reader must pay some attention, even if the context should generally be sufficient to avoid any confusion and un- certainty about the actually used quantity. dW = −dU e = − ′ q dV dW =  Fid  r = ′ q  Eid  r ⎧ ⎨ ⎩ ⎪ , thus, by comparing the two formulae, the difference in electrostatic potential between points P and ′ P is given by dV = −  Eid  r if they are adjacent, or, if the distance between them is finite, it is given by V ′ P ( ) − V P ( ) = −  Eid  r P ′ P ∫ . Electrostatic potential is measured in volt, symbol V, defined as [V] = [J]/[C]. 2.7 Electrostatic Potential of Charge Distributions Each charge which belongs to a system of point-like particles exerts a conservative force.

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  Workshop Physics Activity Guide Module 2

  Mechanics II

  • Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  Two carts can even “explode” apart if you release a compressed spring between them. In this unit we will introduce two new concepts that are useful for studying the interactions just described—work and energy. We start by considering both intuitive and mathematical definitions of the work done on objects. As we will see, the physical definition of work requires that we find the component of a force in the direction of the displacement, which is accomplished using a mathematical procedure known as the “dot product.” We then use Newton’s second law to show that the net work done on an object causes a change in the object’s kinetic energy (it’s energy of motion). Energy turns out to be one of the most powerful and challenging concepts in science, and we will eventually develop a relationship between work and energy that is every bit as powerful as the momentum principle. UNIT 10: WORK AND ENERGY 315 PHYSICAL WORK AND POWER 10.2 THE CONCEPT OF PHYSICAL WORK In this section, we will push and pull on some heavy objects to get a feel for physical work. To do this, you’ll need: • 2 large books or masses • 1 bowling ball • 1 wooden block But first we begin with some predictions. As president of the Load-n-Go Moving Company you need to decide which of three jobs to bid on. All three jobs pay the same amount of money, so you want to choose the job that you think will be the easiest. • Job 1: Lifting one hundred 20-kg boxes a distance of 6 m straight up. • Job 2: Moving one hundred 20-kg boxes a distance of 10 m along a 30 degree incline. • Job 3: Lifting fifty 40-kg boxes a distance of 3 m straight up. 6 m Job 1 Lifting 100 of the 20-kg boxes a distance of 6 m. 3 m Job 3 Lifting 50 of the 40-kg boxes a distance of 3 m. Job 2 Moving 100 of the 20-kg boxes a distance of 10 m along a 30 degree incline on frictionless rollers. 10 m 30° x y Fig. 10.1. A description of jobs to bid on. 10.2.1. Activity: Choosing Your Job Which of the jobs shown in Fig.

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  Engineering Electromagnetics

  Pergamon Unified Engineering Series

  • David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Then the potential of the sphere is, 90 Techniques for the Electrostatic Field and the energy is, (3.67) Which is correct? Actually, they both are. Energy is the ability to do work moving from one point to another. So a reference must be established —a point where by definition the charges have no energy. This would be at infinity, where charges should experience no forces. Referred to infinity, the second result is correct! For this reason we should amend our energy integral to, (3.68) where w(°°) is the reference potential at infinity. Since we will normally take this reference potential to be zero, the former energy expression is usually correct, also. Other Energy Expressions We have seen that the energy stored by a charge distribution located in an electric field is, (3.69) where p and p s are the volume and surface charge distributions, respectively, and u is the potential. Where is this energy located? Does it really matter where it is located? Strictly from an electrostatic point of view, it cannot be determined where the energy is located. We simply know that for a charge to exist in an electric field, or for two charges to exist near one another requires energy. But to specify that the energy is located at one or the other charge, or somewhere in between is impossible in electrostatics. If we look at Maxwell's Equations, however, we can resolve this question to a degree, by showing that the energy is located in space with the electric field. This agrees with our common sense, because we know accelerating charges radiate energy and light (electromagnetic waves) and carries energy with it from one point to another in space (without carrying any charge). We can show that an equivalent expression for the energy in terms of electric fields is, (3.70) This says that wherever an electric field is present in free space, there is an energy density, 5, equal to, (3.71)

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