Electric Field of a Dipole

12 Key excerpts on "Electric Field of a Dipole"

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  Principles of Physics: Extended, International Adaptation

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  3. The magnitude of the electric field set up by an electric dipole at a distant point on the dipole axis (which runs through both particles) can be written in terms of either the product qd or the magnitude p of the dipole moment: E = 1 _____ 2πε 0 qd ___ z 3 = 1 _____ 2πε 0 p ___ z 3 , where z is the distance between the point and the center of the dipole. 4. Because of the 1/z 3 dependence, the field magnitude of an electric dipole decreases more rapidly with distance than the field magnitude of either of the individual charges forming the dipole, which depends on 1/r 2 . LEARNING OBJECTIVES The Electric Field Due to an Electric Dipole Figure 22.3.1 shows the pattern of electric field lines for two particles that have the same charge magnitude q but opposite signs, a very common and important 22.3 The Electric Field Due to a Dipole 635 arrangement known as an electric dipole. The particles are separated by distance d and lie along the dipole axis, an axis of symmetry around which you can imagine rotating the pattern in Fig. 22.3.1. Let’s label that axis as a z axis. Here we restrict our interest to the magnitude and direction of the electric field E → at an arbitrary point P along the dipole axis, at distance z from the dipole’s midpoint. Figure 22.3.2a shows the electric fields set up at P by each particle. The nearer particle with charge +q sets up field E (+) in the positive direction of the z axis (directly away from the particle). The farther particle with charge −q sets up a smaller field E (−) in the negative direction (directly toward the particle). We want the net field at P, as given by Eq. 22.2.3. However, because the field vectors are along the same axis, let’s simply indicate the vector directions with plus and minus signs, as we commonly do with forces along a single axis.

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  . . 22.10 Draw an electric dipole, identifying the charges (sizes and signs), dipole axis, and direction of the electric dipole moment. 22.11 Identify the direction of the electric field at any given point along the dipole axis, including between the charges. 22.12 Outline how the equation for the electric field due to an electric dipole is derived from the equa- tions for the electric field due to the individual charged particles that form the dipole. 22.13 For a single charged particle and an electric dipole, compare the rate at which the electric field magnitude decreases with increase in distance. That is, identify which drops off faster. 22.14 For an electric dipole, apply the relationship between the magnitude p of the dipole moment, the separation d between the charges, and the magnitude q of either of the charges. 22.15 For any distant point along a dipole axis, apply the relationship between the electric field magnitude E, the distance z from the center of the dipole, and either the dipole moment magnitude p or the product of charge magnitude q and charge separation d. Key Ideas ● An electric dipole consists of two particles with charges of equal magnitude q but opposite signs, separated by a small distance d. ● The electric dipole moment p → has magnitude qd and points from the negative charge to the positive charge. ● The magnitude of the electric field set up by an electric dipole at a distant point on the dipole axis (which runs through both particles) can be written in terms of either the product qd or the magnitude p of the dipole moment: E = 1 2πε 0 qd z 3 = 1 2πε 0 p z 3 , where z is the distance between the point and the center of the dipole. ● Because of the 1/z 3 dependence, the field magnitude of an electric dipole decreases more rapidly with dis- tance than the field magnitude of either of the individual charges forming the dipole, which depends on 1/r 2 .

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  Theory and Computation of Electromagnetic Fields

  • Jian-Ming Jin(Author)
  • 2015(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  As a result, in both atoms and nonpolar molecules, the effective center of positive charges will be displaced from the effective center of negative charges, creating a tiny electric dipole in the direction of the electric field. (Here, we assume that the applied field is not strong enough to break the bound electrons loose from the nuclei. In such a case, the matter is often called a dielectric .) In the case of polar molecules, because of the Lorentz force, all the randomly oriented dipoles tend to line up with the applied electric field. When a large number of electric dipoles line in the same direction, the electric fields created by the dipoles add up and these electric fields are in the opposite direction to the applied field, resulting in a weaker total electric field in the medium. To quantify the effect of tiny dipoles, a vector quantity called the dipole moment is defined as 𝓅 = q 𝓁 (1.3.1) where q denotes the charge and 𝓁 denotes the vector pointing from the effective center of the negative charge to that of the positive charge. The sum of dipole moments per unit volume is then P = lim Δ 𝑣 → 0 1 Δ 𝑣 n p i = 1 𝓅 i (1.3.2) where n p denotes the number of dipoles contained in Δ 𝑣 . The dipole moment density P is also called the polarization intensity or polarization vector . When the dipole moment density is uniform, the positive charge of a dipole is completely canceled by the negative charge of the next dipole; hence, there is no net charge in the medium. However, when the dipole moment density is not uniform, the positive charge of 20 BASIC ELECTROMAGNETIC THEORY a dipole cannot be completely canceled by the negative charge of the next dipole, resulting in a net charge at the point and hence a volume charge density. Based on the definition of divergence, this volume charge density is given by 𝜚 e , b = −∇ ⋅ P (1.3.3) where the subscript “b” is used to denote that this is the density of the bound charges.

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  Theory of Electric Polarization

  Dielectrics in Static Fields

  • Bozzano G Luisa(Author)
  • 2012(Publication Date)
  • Elsevier Science

    (Publisher)

  ELECTRIC DIPOLES AND MULTIPOLES 11 The values of molecular dipole moments are usually expressed in Debye units. The Debye unit, abbreviated as D, equals 1 0 1 8 electrostatic units (e.s.u.). The permanent dipole moments of non-symmetrical molecules generally lie between 0.5 and 5D. This order of magnitude can be expected, because the elementary charge e 0 is 4.4 1 0 1 0 e.s.u. and the distances of the charge centres in the molecules amount to about 1 0 9 -1 0 8 cm. Since the most important quantity in the theory of electric polarization, the dipole moment, is generally expressed in Debye units, we prefer for present purposes to use electrostatic units rather than Giorgi's m.k.s. units. For readers accustomed to the m.k.s. system, we may mention that the electrostatic system is based upon a definition of the charge arising from Coulomb's law: for two equal unit charges 1 cm apart, the mutual force is 1 dyne. Thus Coulomb's law can be represented by F = e 1 e 2 /r 2 . The e.s.u. of electric charge is equal to 3.33 1 0 1 0 coulombs. §2. The electric field of an ideal dipole in vacuo Before considering the field of an ideal dipole in vacuo, the general theory of the electrostatic field in vacuo will be briefly summarized.* (a) The potentials and fields due to electric charges According to Coulomb's experimental inverse square law, the force be-tween two charges e and e' with distance r is given by: ee' Υ F = ^~ (1.3) r L r Taking one of the charges, say e as a test charge to measure the effect of the charge e on its surroundings, we arrive at the concept of an electric field produced by e and with a field strength or intensity defined by: Ε = lim— (1.4) e -o e The field strength due to an electric charge at a distance r is then given by: * See for instance J. A. Stratton. 1 12 C. J. F. BOTTCHER e r E = -2 -, (1.5) in which r is expressed in cm, e in electrostatic units and Ε in dynes per charge unit, i.e.

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  Physics, Volume 2

  • David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Figure 26-19a shows a dipole in a uniform electric field (This field is not that of the dipole itself but is produced by an external agent not shown in the figure.) The dipole moment makes an angle  with the direction of the field. We assume the field to be uniform, so that has the same E B p B E B . p B magnitude and direction at the location of  q and  q. The forces on  q and  q therefore have equal magnitudes F  qE but opposite directions, as shown in Fig. 26-19a. The net force on the dipole due to the external field is there- fore zero, but there is a net torque about its center of mass that tends to rotate the dipole to bring into alignment with The torque on each charge is given by   Fr  ; the net torque about the center of the dipole due to the two forces has magnitude (26-25) and its direction is perpendicular to the plane of the page and into the page, as indicated in Fig. 26-19b. We can write Eq. 26-25 as (26-26) Equation 26-26 can be written in vector form as (26-27) which is consistent with the directional relationships for the cross product, as shown by the three vectors in Fig. 26-19b. As is generally the case in dynamics when conservative forces act (the electrostatic force is conservative, as we dis- cuss in Chapter 28), we can represent the system equally well using either force equations or energy equations. Let us therefore consider the work done by the electric field in turning the dipole through an angle . Using the appropriate expression for work in rotational motion (Eq. 11-25), the work done by the external field in turning the dipole from an initial angle  0 to a final angle  is (26-28) W   dW     0  B  d  B     0   d ,  B  p B  E B ,   (qE)d sin   (qd)E sin   pE sin .   F d 2 sin   F d 2 sin   Fd sin , E B . p B 600 Chapter 26 / The Electric Field (b) (a) – + F –F p p E q q d Figure 26-19.

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  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The electric field strength at P can then be determined by dividing D by ε of the medium. Figure 16.17 Determination of an expression for the electric field strength at P due to an electric dipole. The unit vector i is taken along the dipole axis. We now derive an expressions for the electric flux density and field strength at a point at a distance r from the centre of a dipole of length 2 l for the special case in which. Using the cosine rule (Appendix A.3.3) we obtain and From the rule for vector addition we can write and from which we can determine expressions for and as follows. Substituting these results in Equation (16.9) we obtain Expanding each of the terms in brackets in the denominators in a binomial series (Appendix A.9.2) and neglecting terms of order etc. which are small compared to we get Finally, since, we get and hence (16.10) or (16.11) where the quantity p is the electric dipole moment of the dipole which is defined as where n is a unit vector directed along the axis of the dipole from negative to positive (for example, the i direction in Figure 16.17). Torque on an electric dipole in an electric field Consider an electric dipole in a uniform electric field of strength E, as shown in Figure 16.18. The dipole experiences the effect of equal and opposite forces of magnitude qE acting on each pole; that is, a torque (Section 7.3) of, where is the angle between the magnetic axis and the direction of the field. This can be written in vector form as (16.12) where p is the electric dipole moment as defined above. Figure 16.18 Torque on an electric dipole in a uniform electric field. Potential energy of an electric dipole in an electric field Finally in this section, we derive an expression for the potential energy of an electric dipole in an electric field which we assume is uniform over the dimensions of the dipole

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  The Electromagnetic Field

  • Albert Shadowitz(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  3. The force on a dipole A completely separate problem from that in Example 2 is this: find the force exerted on an electric dipole by an external electric field E .

  Figure 6-5 illustrates this case. Suppose we consider the i th component (i = 1.2,3) of the force F * on the positive charge +q of a dipole. Let the i th electric field component at the center of the dipole be E i ; then at the point l where the charge +q is located, this component, by a Taylor expansion, is E i + (∇E i ) (1/2). The higher-order terms have been neglected here by assuming that is small. So, F i + = (+q ) [E ¡ + (∇E i ) · (+1/2)], while similarly, the i th force component on the negative charge is

  The total i th component force on the dipole, itself, will then be

  So

  This was derived for the case of a dipole, with small but finite 2; but the result is unchanged in the limit of a point dipole with the same dipole moment p. Consequently, this is called the dipole force; though, for a real dipole it is only true if the field does not vary too fast with distance. Otherwise one would require higher-order terms in the expansion. The meaning of the expression for F i can be seen more clearly by explicit expansion, say, of the x component.

  FIGURE 6-4

  If E x is a constant then F x is zero. Similarly for the other components. A translational force is exerted on an electric dipole only when the electric field is not constant, but varies from point to point.

  An alternative expression for the force on the dipole may be obtained from a vector identity.In ∇(A · B) = (A · ∇)B + (B · ∇)A + A × (∇ × B) + B X (∇ × A)

  let A = p and B = E .Then

  F = ∇ (p · E ) (E · ∇)p - p × (∇ × E ) - E × (∇ × p)

  The dipole moment p is given here; it does not depend on x , y , or z . Consequently

  (E · ∇ )p = 0 and ∇ × p = 0

  so the electrical force on the dipole is

  F = ∇ (p · E ) - p × (∇ × E )

  But in electrostatics

  ∇ × E = 0

  so

  F = ∇(p · E ) = - ∇(-p · E )

  FIGURE 6-5

  This result is only valid in electrostatics. When E varies with time it will be found subsequently that ∇ X E ≠ 0; then the expression for F also contains the term -p × (∇ × E).

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  Electricity and Magnetism

  • Edward Purcell(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  Clearly each of these charge distributions can be replaced by a point charge at its center, if we are concerned with the field well outside the distribution. "Well outside" means far enough away from the surface so that the actual graininess of the charge distribution doesn't matter, and of course that is some- thing we always have to ignore when we speak of the macroscopic fields. So for present purposes the picture of overlapping spheres of uniform charge density and the description in terms of actual dipoles in a vacuum are equivalent,t and show that the field outside the dis- tribution is the same as that of a single dipole located at the center. The moment of this dipole Po is simply the total polarization in the sphere: A sphere of lined-up molecular dipoles (a) is equivalent to superposed, slightly displaced, spheres of positive (b) and of negative (c) charges. 47r 3 47r 3 Po = Qs = "3 roNqs = "3 roP (35) The quantities Q and s have, separately, no significance and may now be dropped from the discussion. The external field of the polarized sphere is that of a central dipole Po, not only at a great distance from the sphere; it is the pure dipole field right down to the surface, macroscopically speaking. All we had to do to construct Fig. 10.23, a representation of the external field lines, was to block out a circular area from Fig. 10.6. tThis may have been obvious enough, but we have labored the details in this one case to allay any suspicion that the "smooth-charge-ball" picture, which is so different from what we know the interior of a real substance to be like, might be leading us astray. 376 FIGURE 10.23 The field outside a uniformly polarized sphere is exactly the same as that of a dipole located at the center of the sphere. CHAPTBRTEN The internal field is a different matter. Let's look at the electric potential, ip(x, y, z). We know the potential at all points on the spher- ical boundary because we know the external field.

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  Introduction to Electrodynamics

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

    (Publisher)

  C H A P T E R 4 Electric Fields in Matter 4.1 POLARIZATION 4.1.1 Dielectrics In this chapter, we shall study electric fields in matter. Matter, of course, comes in many varieties—solids, liquids, gases, metals, woods, glasses—and these sub- stances do not all respond in the same way to electrostatic fields. Nevertheless, most everyday objects belong (at least, in good approximation) to one of two large classes: conductors and insulators (or dielectrics). We have already talked about conductors; these are substances that contain an “unlimited” supply of charges that are free to move about through the material. In practice, what this ordinarily means is that many of the electrons (one or two per atom, in a typical metal) are not associated with any particular nucleus, but roam around at will. In dielectrics, by contrast, all charges are attached to specific atoms or molecules—they’re on a tight leash, and all they can do is move a bit within the atom or molecule. Such microscopic displacements are not as dramatic as the wholesale rearrangement of charge in a conductor, but their cumulative effects account for the characteris- tic behavior of dielectric materials. There are actually two principal mechanisms by which electric fields can distort the charge distribution of a dielectric atom or molecule: stretching and rotating. In the next two sections I’ll discuss these processes. 4.1.2 Induced Dipoles What happens to a neutral atom when it is placed in an electric field E? Your first guess might well be: “Absolutely nothing—since the atom is not charged, the field has no effect on it.” But that is incorrect. Although the atom as a whole is electrically neutral, there is a positively charged core (the nucleus) and a nega- tively charged electron cloud surrounding it. These two regions of charge within the atom are influenced by the field: the nucleus is pushed in the direction of the field, and the electrons the opposite way.

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  Conceptual Electromagnetics

  • Branislav M. Notaroš(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  The charges are displaced from their equilibrium positions by forces [Eq. (1.3) ] F e1 = Q E and F e2 = −Q E, respectively. The displacement d is very small, on the order of the dimensions of atoms and molecules. The charges Q and −Q are bound in place by atomic and molecular forces and can only shift positions slightly in response to the external field. So, the two charges in an equivalent small dipole cannot separate one from the other and migrate across the material in opposite directions run by the electric field. Hence, these charges are called bound charges (in contrast to free charges). Some dielectrics, such as water, have molecules with a permanent displacement between the centers of the positive and negative charge, so that they act as small electric dipoles even with no applied electric field. According to Figure 2.1, the torques (moments) of forces F e1 and F e2 with respect to the center of the dipole (point O) are T 1 = r 1 × F e1 and T 2 = r 2 × F e2, with r 1 and r 2 denoting the position vectors of Q and −Q with respect to the dipole center. We notice that r 1 − r 2 = d, and thus the resultant torque on the dipole turns out to be T on dipole = T 1 + T 2 = Q (r 1 − r 2) × E = Q d × E ext = p × E. (2.1) The process of making atoms and molecules in a dielectric behave as microscopic electric dipoles and orienting (rotating) the dipoles toward the direction of the external field, by means of the torque in Eq. (2.1), is termed the polarization of the dielectric, and bound charges are sometimes referred to as polarization charges

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  Dielectrophoresis

  Theory, Methodology and Biological Applications

  • Ronald R. Pethig(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  This was used as a mathematical tool by Maxwell, to show that rather than representing fluid flow the lines of force show the direction of the vector field E at any point in space. The concept of a ‘line of electric force’ is thus replaced with the concept of an ‘electric field line’. An indication of the field magnitude is provided by the spacing of these lines. Where the field is strong the lines are bunched together and they are spaced far apart where the field is weak. For the case of a point positive charge, its field radiates out with spherical symmetry and from Equation (3.6) the magnitude and direction of the vector field E are given by: (3.9) where is the unit radial vector. The more general case of field lines produced by two or more charges in three-dimensional space is shown in Figure 3.8. The direction of the electric field vector at any point along a field line is given by the tangent at that point. Apart from the special case of a uniform electric field, the magnitude of the field varies from point to point along a field line. In general, therefore, a vector field E in three-dimensional space has components E x, E y, E z, along the orthogonal x -, y - and z -axes: (3.10) where î, ĵ, and are the unit vectors along the x -, y - and z -axes, respectively. The components E x, E y, and E z may exhibit gradients (i.e., each varying as a function of distance along the x -, y - and z -axes, respectively). This is important for us, because as given by Equation (1.5) of Chapter 1, and in Box 2.4 of Chapter 2, the dielectrophoretic force depends on the gradient of the electric field. This gradient is expressed mathematically as the vector ∇E (variously known as ‘grad’ E or ‘del’ E) where the symbol ∇ functions as a differential operator, such that: (3.11) Figure 3.8 Some of the field lines are drawn between a positive and negative charge. The direction of the electric field vector E at point (e.g., A or B) is given by the tangent to the field line at that point

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  Principles of Induction Logging

  • Avital Kaufman, Yu.A. Dashevsky(Authors)
  • 2003(Publication Date)
  • Elsevier Science

    (Publisher)

  The electric field is located in planes perpendicular to the dipole axis and it has only the component E ϕ. In accord with eq. 2.6 : E ϕ = curl ϕ A * = 1 R (∂ ∂ R (R A θ *) − ∂ A R * ∂ θ) (2.17) It is obvious that: A R * = A z * cos θ A θ * = − A z * sin θ Substituting these expressions into eq. 2.17 after simple transformations we obtain: E ϕ = C 1 R 2 e i k R (1 − i k R) sin θ (2.18) In particular, for a nonconducting medium letting σ = 0 we have: E ϕ = C 1 R 2 sin θ since k = 0. It is obvious that this expression describes the primary vortex electrical field, caused by a change of the primary magnetic field with time, that is: E ϕ (0) = C 1 R 2 sin θ On the other hand, as follows from the previous chapter (section 1.4), we. have: E ϕ (0) = i ω μ M 4 π R 2 sin θ Comparing these last two expressions we define the unknown constant C : C = i ω μ M 4 π (2.19) The vector of the magnetic field has two components: H θ = σ A θ * − grad θ U * H R = σ A R * − grad R U * In accord with eq. 2.16 we. have: grad R U * = M 4 π e i k R (2 i k R 2 − 2 R 3 + k 2 R) cos θ grad θ U * = − M 4 π R 3 e i k R (1 − i k R) sin θ Therefore: H θ = M 4 π R 3 e i k R (− k 2 R 2 + 1 − i k R) sin θ[--=PLGO-SE. PARATOR=--]H R = 2 M 4 π R 3 e i k R (1 − i k R) cos θ Finally, we obtain the following expressions for complex amplitudes of the electromagnetic field of the magnetic. dipole: E ϕ = i ω μ M 4 π R 2 e i k R (1 − i k R) sin θ (2.20) H R = 2 M 4 π R 3 e i k R (1 − i k R) cos θ (2.21) H θ = M 4 π R 3 e i k R (1 − i k R − k 2 R 2) sin θ (2.22) Proceeding from these equations we will investigate the behavior of the electromotive force induced in the receiver of the. two coil induction probe as well as the main features of the distribution of induced currents. Suppose that a receiver coil is significantly smaller than the length of the induction probe. In other words, it will be assumed that all turns of the coil have the same area and they are located at some distance from the transmitter coil

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