7 Key excerpts on "Divergence of Electrostatic Field"

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  eBook - ePub

  The Electrical Engineering Handbook

  • Wai Kai Chen(Author)
  • 2004(Publication Date)
  • Academic Press

    (Publisher)

  Section 2.2.3 .

  2.2.1 D Fields of Charge Distributions Using Gauss’s Law

  The following derivations require the calculus of vector differential operators. Most textbooks in electromagnetic theory contain the relevant theorems and their application to electromagnetic fields. A particularly complete and concise presentation of the same can be found in Chapter 2 of the textbook by Jefimenko (1996). The book by Schey (1996) is also recommended. Helmholtz’s theorem states that any vector field that is continuous and regular at infinity can be completely specified by its divergence and curl. Thus, including the mediation of the permittivity function, it is clear that the electrostatic field is completely defined by equations 2.2 . Integrating both sides of equation 2.2c over all of space and applying the divergence theorem yields for the left-hand side:

  (2.4)

  Integration of the right-hand side over all of space encompasses the volume containing the charge density and simply gives the total charge enclosed. The result is Gauss’s Law stating that the total electric flux crossing any surface bounding the charge is equal to the total charge enclosed in that surface, clearly a conservation law:

  (2.5)

  where S is the bounding surface of the integration volume V and where the vector differential surface element is given by the differential surface element, dS , multiplied by the unit vector normal, n , to the surface at the point of integration as Figure 2.1 illustrates.

  FIGURE 2.1 Gauss’s Law Connects the Total Source of Flux Contained Inside the Volume V to the Flux Density that Exits its Bounding Surface S .

  With equation 2.5 , the electric flux density, or D field, from highly symmetric charge distributions can be obtained. The method of approach is to guess at the form of the field and pick a bounding surface over which the D field is constant and perpendicular to the surface, thus allowing it to be pulled out of the integral on the left-hand side of equation 2.5 . The bounding surface is known as a Gaussian surface.

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  eBook - ePub

  Electrical Engineering

  Fundamentals

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

    (Publisher)

  Paraelectric polarisation: Insulating materials are already organised as dipoles. However, without an outer electric field, the electric effects cancel each other out due to the disordered thermal movement (a). If an outer field is applied, the dipoles align themselves according to their polarities (b).

  4.5  The electric displacement flux Ψ

  Electric fields are the sum of all field lines. The fields occur due to the redistribution of charges; the sum of all field lines therefore is called displacement flux Ψ , which corresponds to the amount of separated charges. The following applies:

  Ψ = Q

  .

  The displacement flux is the description of the electric charge with regard to the field. The number of field lines per perpendicular surface unit varies depending on the development of the field. The displacement flux per perpendicular surface unit is called displacement flux density D . The following applies:

  D ⇀

  =

  Ψ

  A ⇀

  . It is a vector and behaves proportionally to the electric field strength. In a vacuum, the relation:

  D =

  ε 0

  ⋅ E

  applies, in other substances, it is

  D =

  ε 0

  ⋅

  ε r

  ⋅ E

  . Thereby,

  ε 0

  is the electric constant and

  ε r

  the material-dependent permittivity.

  Figure 4.9: The electric displacement flux Ψ is the quantity of electric charge Q redistributed by an electric field.

  4.6  Dielectric

  Electrically insulating materials in components where strong electric fields occur are usually called dielectrics (e.g. cable insulation, capacitors). The dielectric strength

  E d

  of a dielectric is the maximum permissible electric field strength at which the material is not destroyed. It is indicated in

  k V

  m m

  .

  4.6.1  Permittivity ε (formerly known as dielectric constant)

  Permittivity (permeability of a material for electric fields)

  Table 4.1: Selection of technically used dielectrics.

  Material	ε r	E d   i n   k V / m m
  Air (normal pressure)	1	2.1
  Water (distilled)	80	–
  Natural mica	6…8	30…70
  China	5 … 6	35
  Polyethylene (PE)	2.3	60…90
  Polystyrene (PS)	2.3…4.2	35
  Epoxy resin	3.7…4.2	35
  Silicone rubber	2.5	20…30

  4.7  The capacitor

  Capacitors in their basic form (see Figure 4.10

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  Basics of Electromagnetics and Transmission Lines

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

    (Publisher)

  2 = 2 cm from it?

  (a)  E2 = 1 V/m

  (b)  E2 = 4 V/m

  (c)  E2 = 2 V/m

  (d)  E2 = ½ V/m

  3.    The intensity of the field due to a line charge p

  L1

  at a distance r1 = 1 cm away from it is E1 = 1 V/m. What is the intensity E2 of the field of the line charge p

  L2

  = 4 at a distance r2 = 2 cm from it?

  (a)  E2 = 1 V/m

  (b)  E2 = 4 V/m

  (c)  E2 = 2 V/m

  (d)  E2 = ½ V/m

  4.    Charge Q is uniformly distributed in a sphere of radius a1 . How is the charge density going to change if this same charge is now occupying a sphere of radius a2 = a1 /4 ?

  (a)  It will increase 4 times (b)  It will increase 64 times (c)  It will increase 16 times (d)  It will increase 2 times

  5.    A line charge p

  L

  = 5 × 103 C/m is located at (x, y) = (0, 0), and is along the z-axis. Calculate the surface charge density p

  s

  (p

  s

  > 0) and the location x

  p

  (x

  p

  > 0) of an infinite planar charge distributed on the plane at x = x

  p

  , so that the total field at the point P (0. 5 × 10−3 , 0) m, is zero.

  (a)  ρ

  s

  = 1/(2π) C/m2 , x

  p

  = 5 × 10−3 m

  (b)  ρ

  s

  = 1/(2π) C/m2 , ∀x

  p

  (c)  ρ

  s

  = 1/π C/m2 , x

  p

  = 10 × 10−3 m

  (d)  ρs = 1/π C/m2 , ∀x

  p

  6.    The volume charge density associated with the electric displacement vector in spherical coordinates (sinθsinϕa

  r

  + cosθsinϕaϕ + cos ϕa

  ϕ

  ) is

  (a)  0 (b)  1 (c)  Not compatible (d)  sinθ

  7.    The divergence theorem

  (a)  Relates a line integral to a surface integral (b)  Holds for specific vector fields only (c)  Works only for open surfaces (d)  Relates a surface integral to a volume integral

  8.    The flux of a vector quantity crossing a closed surface

  (a)  is always zero (b)  is related to the quantity’s component normal to the surface (c)  is related to the quantity’s component tangential to the surface (d)  is not related in any way to the divergence of that vector quantity

  9.    The flux produced by a given set of fixed charges enclosed in a given closed region is

  (a)  Dependent on the surface shape of the region, but not the volume

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  eBook - ePub

  Electromagnetic Waves

  • Carlo G. Someda(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  1.1 ).

  We shall not overlook the case where one or more of the above-listed quantities is not regular throughout its domain of definition. Indeed, we will see, in Section 1.6 , that some of the electromagnetic vectors can be discontinuous across surfaces where the materials which occupy their domain of definition have step discontinuities. In order to “justify” Maxwell’s equations and the charge continuity equation in such circumstances, we will accept, in our theory, material discontinuities only if they satisfy the following two requirements, which agree fully with physical intuition:

  1. the number of discontinuities over any finite interval in space is finite;
  2. each discontinuity is of the first type, i.e., it can be interpreted as the limit of a continuous transition (distributed over a characteristic length δ , and having derivatives with respect to spatial coordinates up to the order one needs) as δ tends to zero (see Figure 1.1 ).

  Under these assumptions, the curl and divergence vector operators, defined in Appendix A, and used as described above, lead always to vector functions that admit a right limit and a left limit everywhere. Where these two limits do not coincide, an integration (over a surface, in the case of a curl, over a volume, in that of a divergence) leads to a relation between continuous quantities , as required by the macroscopic laws Eqs. (1.1 ), (1.3 ) and (1.4 ).

  The subject of discontinuities is a delicate topic. We will come back to it, shortly, in Section 1.5 .

  1.3 Constitutive relations

  In Section 1.1 , we anticipated that the vectors

  B ∼

  and

  H ∼

  are not independent of each other. As the reader should know from previous studies of electrostatic and magnetostatic phenomena, in general the vectors defined so far are related not only by Maxwell’s equations and the charge continuity equation, but also by other relations, which express properties of the medium where these vectors are defined. They are referred to as the constitutive relations

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  eBook - ePub

  Engineering Electrodynamics

  Electric Machine, Transformer, and Power Equipment Design

  • Janusz Turowski, Marek Turowski(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  x located inside a volume V ([2.10], p. 59).

  Electrostatics and magnetostatics are the particular and simplest cases of electrodynamics. Electrostatic and magnetostatic phenomenon are ruled by the same Maxwell’s equations, which at the assumption of DC fields (∂ D /∂t = ∂ B /∂t = 0) outside any sources (ρ = 0) and currents (J = 0) in the region, for immovable media, take the form:

  • For electrostatics:
    curl E = 0 ,

    div D = ρ

    ,

    D

    = ε E

    ⁢ ( 2.23 )
  • For magnetostatics:
    curl H = 0 ,

    div B = 0

    ,

    B

    = μ H

    ⁢ ( 2.24 )

  As one can see, these fields can exist and be investigated fully independent of each other. The equations curl E = 0 and curl H = 0 confirm that these fields are irrotational . Such fields are potentional , which means for their description one can introduce scalar functions of position V (x , y , z ) and V m (x , y , z ) called electric and magnetic potentials , respectively. These potentials, per the principle of identity of vector calculus (curl grad V ≡ 0), fulfill the relations:

  E = - grad V

  ⁢ ( 2.23a )

  H = - grad

  V m

  ⁢ ( 2.24a )

  The next particular case of electrodynamics is electroflow field, in which we take into account the presence of currents steady in time. Assuming (∂ D /∂t = ∂ B /∂t = 0) and J ≠ 0, we obtain the fundamental equations of electroflow field (for immovable, conducting bodies):

  curl H = J ,

  curl E = 0

  d i v B = 0 ,

  d i v D = 0

  J = σ

  ( E +

  E extern

  ) ,

  div J = 0

  ⁢ ( 2.25 )

  The equations div J = 0 and curl E = 0 are differential equivalents of the 1st and 2nd Kirchhoff’s Equations.

  2.2 Formulation and Methods of Solution of Field Differential Equations

  The process of solution for electrodynamic problems comprises (Figure 1.1

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  High Voltage Engineering Fundamentals

  • John Kuffel, Peter Kuffel(Authors)
  • 2000(Publication Date)
  • Newnes

    (Publisher)

  27)

  Let us consider a steady state electrostatic field within a dielectric material whose conductivity may be neglected and whose permittivity may be dependent upon the direction of the field strength E (anisotropic material) or not (isotropic dielectric). Then as no space charge should be present or accumulated, the potentials would be excited from boundaries (metal electrodes) between which the dielectric material is placed. Assuming a Cartesian coordinate system, for such a Laplacian field, the electrical energy W stored within the whole volume R of the region under consideration is

  (4.50)

  εx , εy

  and

  εz

  would be anisotropic permittivity coefficients, and it should be noted that even in an isotropic material with

  εx = εy = εz = ε

  , the absolute values of ε may change at boundaries between different dielectric materials. The reader may easily verify from any small volume element d V = (dx dy dz) that the expressions (ε∇2 ϕ/2) within eqn (4.50) are energy densities per unit volumes dV .

  Furthermore, it is assumed that the potential distribution does not change in the z -direction, i.e. a two-dimensional case. Figure 4.28 displays the situation for which the field space is reduced from the volume R to the area A limited by boundaries with given potentials

  ϕa

  and

  ϕb

  , (Dirichlet boundaries). The dielectric may be subdivided into two parts, I and II, indicated by the dashed interface, for which the boundary condition is well known (see section 4.3 ), if no free charges are built up at the interface. The total stored energy within this area-limited system is now given according to eqn (4.50)

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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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