Bound Charge

6 Key excerpts on "Bound Charge"

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  Electricity and Magnetism

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

    (Publisher)

  Over the former we have some degree of control-charge can be added to or removed from an object, such as the plate of a capacitor. This is often calledfree charge. The other charges, which are integral parts of the atoms or molecules of the dielectric, are usually called Bound Charge. Structural charge might be a better name. These charges are not mobile but more or less elastically bound, contributing, by their slight displacement, to the polarization. One can devise a vector quantity which is related by something like Gauss' law to the free charge only. In the system we have just examined, a point charge Q immersed in a dielectric, the vector tE has this property. That is, I tE . da, taken over some closed surface S, equals 41l'q if S encloses Q, and zero if it does not. By superposition, this must hold for any collection of free charges described by a free- charge density Pfree(X, y, z) in an infinite homogeneous dielectric medium: Is tE . da = 41l' Iv Pfree dv (48) where V is the volume enclosed by the surface S. An integral relation like this implies a "local" relation between the divergence of the vector field tE and the free charge density: (49) Since t has been assumed to be constant throughout the medium, Eq. 49 tells us nothing new. However, it can help us to isolate the role of the Bound Charge. In any system whatever, the fundamental relation between electric field E and total charge density Pfree + Pbound remains valid: div E = 41l'(Pfrcc + Pbound) (50) From Eqs. 49 and 50 it follows that div (t - l)E = -41l'Pbound (51) According to Eq. 34, (t - I)E = 41l'P, so Eq. 51 implies that div P = - Pbound (52) EL.CTRIC FIELDS IN MATTER Equation 52 states a local relation. It cannot depend on condi- tions elsewhere in the system, nor on how the particular arrangement of Bound Charges is maintained.

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  A Concise Handbook of Mathematics, Physics, and Engineering Sciences

  • Andrei D. Polyanin, Alexei Chernoutsan(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  This charge equals Δ q = n ( Δ s 〈 l 〉 cos θ ) q 0 = P n Δ s = ( P ⋅ Δ s ) (P3. 5 . 1 . 1 ) where Δ s 〈 l 〉 cos θ is the volume of a thin cylindrical layer whose molecules cross the area (Fig. P3.11). Integrating over any closed surface yields the net Bound Charge displaced 482 E LECTRODYNAMICS from the enclosed volume (the net Bound Charge enclosed within the surface will have the opposite sign): q b enc = – contintegraldisplay P ⋅ d s or div P = – ρ b , (P3. 5 . 1 . 2 ) where d s = n ds is the surface element vector directed along the normal toward the sur-rounding space and ρ b is the volume density of the Bound Charge. The surface density of the Bound Charge of (P3.5.1.1) is given by σ b = P n . (P3. 5 . 1 . 3 ) ◮ Electric displacement. The macroscopic electric field, E ( r ), in a polarized dielectric is determined by averaging the microscopic field, E micro (highly varying at interatomic distances), over a small volume containing sufficiently many molecules: E ( r ) = 〈 E micro 〉 . The total electric field is determined by both Bound Charges and free charges brought to the dielectric from outside: contintegraldisplay E ⋅ d s = 1 ε 0 ( q f enc + q b enc ). (P3. 5 . 1 . 4 ) Since the Bound Charge distribution is unknown in advance, it is convenient to introduce a vector quantity D = ε 0 E + P , (P3. 5 . 1 . 5 ) which is called the electric displacement field or just the electric displacement . It follows from equations (P3.5.1.2) and (P3.5.1.4) that the electric displacement is determined by the free charges only: contintegraldisplay D ⋅ d s = q f enc . (P3. 5 . 1 . 6 ) This equation is known as Gauss’s law for the electric displacement . In differential form, this law reads div D = ρ f . ◮ Isotropic dielectric. The polarization density P at a given point in a dielectric is determined by the external electric field E . For not too strong fields, P is linearly dependent on E .

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  Basic Electromagnetic Theory

  • James Babington(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  The experimental apparatus Faraday used in 1836 to quantitatively study the effect of dielectrics in between capacitor plates. In this version, the spherical capacitor plates geometry removed any doubts about the particular geometric configurations (e.g. edge effects).

  From this experiment we establish the fact that the charge density has changed on the plates. So how should we interpret this? If the charge density on the plates has changed then the charge density in the insulator must also have changed to produce this effect. But it can’t be the simple addition or subtraction of charge because this would lead to a violation of the conservation of charge. It would give an observable net Coulomb like force outside the capacitor. There are obviously charges inside the insulator, but they must be in the form of bound states - groups of charge that form a stable unit. When the insulator is placed between the plates, the electric field due to the potential difference must cause the charges in the insulator to be redistributed. In terms of the bound state they are displaced with respect to one another, but relax back to their original configuration when they are taken out from the capacitor plates. This then is the picture we have of the mechanism by which the interaction takes place.

  Let us try and capture this in terms of the field equations [8 ]. Consider Equation (4.53 ) (Maxwell I) where a charge density is a source for an electric field. We can actually think about separating the charge density into different contributions, depending on a length scale and whether it is free charge or is part of a bound state. If we do this then

  (7.1)

  By free charge we mean charge that can run around in the material subject only to resistive forces (collisions, but what really are collisions?) that will eventually bring it to a stop. For a bound state charge, a phenomenological model could be a pair of equal and opposite charges that have an harmonic potential binding them together (a spring). This is perhaps the simplest example of a bound state charge distribution. Note there that the spring constant would have to be introduced by hand - it is a phenomenological parameter that isn’t calculated but would have to be measured. In general, however, the distribution of charge in a bound state will be more complicated and so we now address this point.

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  College Physics

  • Paul Peter Urone, Roger Hinrichs(Authors)
  • 2012(Publication Date)
  • Openstax

    (Publisher)

  The law of conservation of charge is absolute—it has never been observed to be violated. Charge, then, is a special physical quantity, joining a very short list of other quantities in nature that are always conserved. Other conserved quantities include energy, momentum, and angular momentum. PhET Explorations: Balloons and Static Electricity Why does a balloon stick to your sweater? Rub a balloon on a sweater, then let go of the balloon and it flies over and sticks to the sweater. View the charges in the sweater, balloons, and the wall. Figure 18.10 Balloons and Static Electricity (http://cnx.org/content/m42300/1.5/balloons_en.jar) 18.2 Conductors and Insulators Figure 18.11 This power adapter uses metal wires and connectors to conduct electricity from the wall socket to a laptop computer. The conducting wires allow electrons to move freely through the cables, which are shielded by rubber and plastic. These materials act as insulators that don’t allow electric charge to escape outward. (credit: Evan-Amos, Wikimedia Commons) Some substances, such as metals and salty water, allow charges to move through them with relative ease. Some of the electrons in metals and similar conductors are not bound to individual atoms or sites in the material. These free electrons can Chapter 18 | Electric Charge and Electric Field 699 move through the material much as air moves through loose sand. Any substance that has free electrons and allows charge to move relatively freely through it is called a conductor. The moving electrons may collide with fixed atoms and molecules, losing some energy, but they can move in a conductor. Superconductors allow the movement of charge without any loss of energy. Salty water and other similar conducting materials contain free ions that can move through them. An ion is an atom or molecule having a positive or negative (nonzero) total charge. In other words, the total number of electrons is not equal to the total number of protons.

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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)

  So the an electric field, atom now appears as a nucleus surrounded by a cloud of negative charge displaced from center by a small amount. Thus each atom appears to be a small atomic dipole. The displacement of the nuclei and electrons of millions of atoms will produce an additional electric field opposing the original field. This is the polarization field, P, and is defined as, (4.1) where p is the total dipole moment in a volume element, AK. That is to say the polarization, P, is the dipole moment per unit volume, and is a vector field which varies from point to point in a material. We shall presently discuss in more detail how this comes about on an atomic scale, but first we relate the macroscopic impact of polarization on the field equations. To conceptually link this newly defined polarization vector, P, to our field vectors, E and D, consider a thin layer of dielectric material with an electric field passing through it (see Fig. 4-3). This field, E d , induces polarization, P, or if you Fig. 4-3. A thin layer of dielectric. prefer dipole moment, p. The amount of polarization, P, depends on the material. Thus it is impossible without knowledge of the material (its density, atomic structure, molecules, valence, etc.), to determine P from E. Within this material all charges experience a force which displaces them a distance, d. At the surface, however, this displacement cannot occur as the material will hold its charges in. 118 Dielectric Materials So an excess surface charge (Bound Charge) is deposited on the surface in the amount, (positive charge density). The total dipole moment, p, on the other hand is, p = d • (positive charge). Comparing equations, (4.2) from the definition of P. The opposite charge must appear on the other surface, i.e., These two layers of bound surface charge cause electric fields within the dielec-tric layer which are, (4.3) and the induced electric field, E P , is directed opposite to the external field, E e .

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  Introduction to Electrodynamics

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

    (Publisher)

  The capacitor is connected to a battery of voltage V . Find all the Bound Charge, and check that the total is zero. Problem 4.35 A point charge q is imbedded at the center of a sphere of linear dielectric material (with susceptibility χ e and radius R). Find the electric field, the polarization, and the Bound Charge densities, ρ b and σ b . What is the total Bound Charge on the surface? Where is the compensating negative Bound Charge located? Problem 4.36 At the interface between one linear dielectric and another, the electric field lines bend (see Fig. 4.34). Show that tan θ 2 / tan θ 1 =  2 / 1 , (4.68) assuming there is no free charge at the boundary. [Comment: Eq. 4.68 is reminiscent of Snell’s law in optics. Would a convex “lens” of dielectric material tend to “focus,” or “defocus,” the electric field?] 21 This charming paradox was suggested by K. Brownstein. 22 Interestingly, it can be done with oscillating fields. See K. T. McDonald, Am. J. Phys. 68, 486 (2000). 4.4 Linear Dielectrics 207 θ 2 θ 1 E 2 E 1  1  2 FIGURE 4.34 Problem 4.37 A point dipole p is imbedded at the center of a sphere of linear ! dielectric material (with radius R and dielectric constant  r ). Find the electric po- tential inside and outside the sphere.  Answer: p cos θ 4πr 2  1 + 2 r 3 R 3 ( r − 1) ( r + 2)  , (r ≤ R); p cos θ 4π 0 r 2  3  r + 2  , (r ≥ R)  Problem 4.38 Prove the following uniqueness theorem: A volume V contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the potential is specified on the bound- aries S of V (V = 0 at infinity would be suitable) then the potential throughout V is uniquely determined. [Hint: Integrate ∇ · (V 3 D 3 ) over V .] V 0 R FIGURE 4.35 Problem 4.39 A conducting sphere at potential V 0 is half embedded in linear dielectric material of susceptibility χ e , which occupies the region z < 0 (Fig. 4.35).

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