Boundary Conditions for Electromagnetic Fields

8 Key excerpts on "Boundary Conditions for Electromagnetic Fields"

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  Electromagnetic Shielding

  Theory and Applications

  • Salvatore Celozzi, Rodolfo Araneo, Paolo Burghignoli, Giampiero Lovat(Authors)
  • 2022(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  1.6 Initial Conditions, Boundary Conditions, and Causality As was noted earlier, Maxwell’s equations together with the constitutive relations represent a set of partial differential equations. However, it is well known that 1.7 Poynting’s Theorem and Energy Considerations 13 in order to obtain a solution for this set of equations, both initial and boundary conditions must be specified. The initial conditions are represented by the constraints that the EM field must satisfy at a given time, while boundary conditions are, in general, constraints that the EM field must satisfy over certain surfaces of the three-dimensional space, usually surfaces that separate regions of space filled with different materials. In these cases the boundary conditions coincide with the jump conditions illustrated in Section 1.5. Other important examples of boundary conditions that can easily be formulated in the frequency domain are the impedance boundary condition and radiation condition at infinity. The impedance boundary condition (also known as the Leontovich condition) relates the component E t of the electric field tangential to a surface S with the magnetic field as E t = Z S ( u n × H ) , (1.32) where Z S (surface impedance) is a complex scalar quantity. The radiation con- dition at infinity (also known as the Sommerfeld radiation condition in scalar radiation problems or the Silver–Müller radiation condition in vector radiation problems) postulates that in free space, in the absence of sources at infinity, there results lim r→+∞ r [ E − 𝜂 0 ( H × u r )] = 𝟎. (1.33) As concerns transient fields, physical grounds require that all the fields satisfy the law of causality. Two different causality conditions can be defined, i.e., a weak and a strong causality condition. The weak causality condition states that all fields have to be zero for t ≤ t 0 if the sources are zero for t ≤ t 0 . In other words, an out- put does not exist until an input is applied.

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  Electromagnetic Radiation, Scattering, and Diffraction

  • Prabhakar H. Pathak, Robert J. Burkholder(Authors)
  • 2021(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  CHAPTER 2 EM BOUNDARY AND RADIATION CONDITIONS 2.1 EM Field Behavior Across a Boundary Surface At macroscopic scales, the boundary surface between two different media is sharply defined. As a result, the macroscopic EM fields can exhibit a discontinuous behavior across such a boundary. Also, EM fields can become discontinuous across a boundary defined by some impressed surface source distribution of current, or charge, whether or not the medium on either side of the surface source distribution is dissimilar. Furthermore, it may so happen in the method of solution that different EM field expansions are chosen for representing the EM fields within different regions into which the problem configuration space is divided for convenience. In such instances, the EM fields have to be properly related across the artificial boundaries created by the analytical formulation of the solution. Therefore, it becomes necessary to develop boundary conditions that serve to relate the behavior of EM fields across boundary surfaces which may arise from the interface between two dissimilar media and/or surface sources, or from analytical formulations where different field representations are joined across mathematical surfaces. These boundary conditions are developed below. Consider a tiny, mathematical pillbox shape consisting of a right circular cylindrical surface of volume V with end caps as shown in Figure 2.1, where V is located with its center on the locally plane but otherwise smooth and curved boundary surface S b , which divides the surrounding space into regions 1 and 2 that, in general, can exhibit different electrical properties, respectively, on either side of S b .

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  Microwave Engineering

  • David M. Pozar(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  We will also see that the magnetic wall boundary condition is analogous to the relations between the voltage and current at the end of an open-circuited transmission line, while the electric wall boundary condition is analogous to the voltage and current at the end of a short-circuited transmission line. The magnetic wall condition, then, provides a degree of completeness in our formulation of boundary conditions and is a useful approximation in several cases of practical interest. The fields at a magnetic wall satisfy the following conditions: ˆ n · ¯ D = 0, (1.40a) ˆ n · ¯ B = 0, (1.40b) ˆ n × ¯ E = − ¯ M s , (1.40c) ˆ n × ¯ H = 0, (1.40d) where ˆ n is the normal unit vector pointing out of the magnetic wall region. The Radiation Condition When dealing with problems that have one or more infinite boundaries, such as plane waves in an infinite medium, or infinitely long transmission lines, a condition on the fields at infinity must be enforced. This boundary condition is known as the radiation condition and is essentially a statement of energy conservation. It states that, at an infinite distance from a source, the fields must either be vanishingly small (i.e., zero) or propagating in an outward direction. This result can easily be seen by allowing the infinite medium to contain a small loss factor (as any physical medium would have). Incoming waves (from infinity) of finite amplitude would then require an infinite source at infinity and so are disallowed. 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS The Helmholtz Equation In a source-free, linear, isotropic, homogeneous region, Maxwell’s curl equations in phasor form are ∇ × ¯ E = − j ωµ ¯ H , (1.41a) ∇ × ¯ H = j ω ¯ E , (1.41b) and constitute two equations for the two unknowns, ¯ E and ¯ H . As such, they can be solved for either ¯ E or ¯ H . Taking the curl of (1.41a) and using (1.41b) gives ∇ × ∇ × ¯ E = − j ωµ∇ × ¯ H = ω 2 µ ¯ E , 16 Chapter 1: Electromagnetic Theory which is an equation for ¯ E .

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  Essentials of Computational Electromagnetics

  • Xin-Qing Sheng, Wei Song(Authors)
  • 2012(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Sommerfeld radiation condition . Actually in electromagnetics, in order to simplify the problem, there are many other boundary conditions derived according to the physical characteristics of the problem. For instance, the surface of the ocean or the ground is approximated by the impedance boundary conditions. Local absorbing boundary conditions can be derived according to either the behavior of the scattered field in a scattering problem, or the transmission modes of waveguides in a propagation problem. Discussion of these conditions is beyond the scope of this text. The interested reader is referred to [1] for more in-depth discussion on impedance boundary conditions, Sections 11.7–11.9 in [2] for more information on local absorbing boundary condition, and [3] for more details on absorbing boundary conditions based on waveguide modes.

  1.1.4 Maxwell's Equations in the Frequency Domain

  In the above three subsections we have introduced the Maxwell's equations, the constitutive relations, and the boundary conditions required to uniquely determine the EM fields in a given domain. Actually, very often we are only interested in the EM field or wave at a single frequency. Or to simplify the problem, we first study the time-harmonic (sinusoidal) EM field behavior, and then obtain the time-varying field through inverse Fourier transformation (IFT). For a time-harmonic EM field, the field can be expressed by the product of a time-independent complex vector and a time-dependent phasor , where is the angular frequency. Thus, (1.9) and (1.10) can be written in the frequency (or phasor) domain as

  (1.16)

  (1.17)

  In the above we have omitted the phasor . In order to clarify the physical meaning of the above equations, displacement electric and magnetic currents are included in the concept of generalized electric and magnetic currents. Under this concept, the induced electric and magnetic current densities are expressed as:

  (1.18)

  (1.19)

  where and

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  Geophysical Electromagnetic Theory and Methods

  • Michael S Zhdanov, Michael S. Zhdanov(Authors)
  • 2009(Publication Date)
  • Elsevier Science

    (Publisher)

  However, we can formulate boundary conditions for an important and commonly met case, that in which a surface S separates two media identified by the indices (i) and (i + 1). On either side of this surface, the properties of the two media are constant, or vary spatially so slowly that within some small region about any arbitrary point the properties can be considered to be constant. We designate as n the unit vector normal to a point on the surface S directed from medium (i) to medium (i + 1), as shown in Figure 3.3. We also assume that there are no extrinsic currents or charges on the surface. We accept the condition of conjugate vectors for the electromagnetic field on the surface S. In so doing, we replace the surface S separating medium (i) from medium (i + 1) with a very thin transitional layer with a thickness Δ £ in which the parameters ∈, μ, and σ vary rapidly but continuously, as in Figure 3.3b. In such a layer, Maxwell’s equations are still valid. Figure 3.3 Formulation of the boundary conditions for electromagnetic field: the surface S separating medium (i) from medium (i + 1) is replaced with a very thin transitional layer with a thickness Δ £ in which the parameters ∈, μ, and σ vary rapidly but continuously. Using Maxwell’s equations, we determine the relationship between field components at the inner and outer edges of the transitional layer and then take a limit as Δ £ → 0. So doing, we find the condition for continuity. In the transitional layer, we form a rectangular cylinder V of a small cross section Δ S (Figure 3.3a) and examine the flux of the vector field B across the surface of the cylinder Σ. From Gauss’s theorem, and on the strength of equation (3.3), this flux is: (3.51) where V is the volume of the cylinder and n Σ is the unit vector normal to the surface Σ. Inasmuch as Δ S is small, we can assume that B is constant over the cylinder

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  Balanis' Advanced Engineering Electromagnetics

  • Constantine A. Balanis(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  A summary of all the boundary conditions for time-harmonic fields, referring to Figure 1-4, is found in Table 1-5. In addition to the boundary conditions found in Table 1-5, an additional boundary condition on the tangential components of the electric field is often used along an interface when one of the two media is a very good conductor (material that possesses large but finite conductivity). 23 TABLE 1-4 Instantaneous and time-harmonic forms of Maxwell’s equations and continuity equation in differential and integral forms Instantaneous Time harmonic Differential form i t ∂ ∇× = − − ∂ @ % } ∇× = − − E M B i j ω i c t ∂ ∇× = + + ∂ $ * ) ) ∇× = + + H J J D i c j ω υ ∇⋅ = e $ q ∇⋅ = D q eυ υ ∇⋅ = m @ q ∇⋅ = B q mυ υ ∂ ∇⋅ = − ∂ e ic t ) q ∇⋅ = − J ic e j q ω υ Integral form i S C S d d d t ∂ ⋅ = − ⋅ − ⋅ ∂ ∫∫ ∫∫ ∫ s s  % } @  E M s B s ⋅ = − ⋅ − ⋅ ∫ ∫∫ ∫∫ d d j d C i S S   ω i c S S C S d d d d t ∂ ⋅ = ⋅ + ⋅ + ⋅ ∂ ∫∫ ∫∫ ∫∫ ∫ s s s  * ) ) $  H J s J s D s ⋅ = ⋅ + ⋅ + ⋅ ∫ ∫∫ ∫∫ ∫∫ d d d j d C i S c S S   ω ⋅ = ∫∫ e S d s $ 4  D s ⋅ = ∫∫ d Q e S  m S d ⋅ = ∫∫ s  @ 4 B s ⋅ = ∫∫ d Q m S  e ic S d t ∂ ⋅ = − ∂ ∫∫ s  ) 4 J s ic e S d j Q ⋅ =− ∫∫ ω  24 TIME-VARYING AND TIME-HARMONIC ELECTROMAGNETIC FIELDS This is illustrated in Figure 1-6 where it is assumed that medium 1 is a very good conductor whose surface, as will be shown in Section 4.3.1, exhibits a surface impedance Z s (ohms) given, approximately, by (4-42) or Z R jX j (1 ) 2 s s s 1 1 ωm σ = + = + (1-62) with equal real and imaginary (inductive) parts ( 1 σ is the conductivity of the conductor). At the surface there exists a linear current density J s (A/m) related to the tangential magnetic field in medium 2 by J n H s  ˆ × 2 (1-63) Since the conductivity is finite (although large), the most intense current density resides at the surface, and it diminishes (in an exponential form) as the observations are made deeper into the conductor. This is demonstrated in Example 5.7 of Section 5.4.1.

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  Classical Electrodynamics

  • John David Jackson(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  If the fields E and B from two source distri- butions are the same at a given point in space, the force acting on a test charge or current at that point will be the same, regardless of how different the source distributions are. This gives E and B in (I.3) meaning in their own right, independent of the sources. Second, electromagnetic fields can exist in regions of space where there are no sources. They can carry energy, momentum, and angular momentum and so have an existence totally independent of charges and currents. In fact, though there are recurring attempts to eliminate explicit reference to the fields in favor of action-at-a-distance descriptions of the interaction of charged particles, the concept of the electromagnetic field is one of the most fruitful ideas of physics, both classically and quantum mechanically. The concept of E and B as ordinary fields is a classical notion. It can be thought of as the classi- cal limit (limit of large quantum numbers) of a quantum-mechanical description in terms of real or virtual photons. In the domain of macroscopic phenomena and even some atomic phenomena, the discrete photon aspect of the electromagnetic field can usually be ignored or at least glossed over. For example, 1 meter from a 100-watt light bulb, the root mean square electric field is of the order of 50 V/m and there are of the order of 10 15 visible photons/cm 2  s. Similarly, an isotropic FM antenna with a power of 100 watts at 10 8 Hz produces an rms electric field of only 0.5 mV/m at a distance of 100 kilometers, but this still corresponds to a flux of 10 12 photons/cm 2  s, or about 10 9 photons in a volume of 1 wavelength cubed (27 m 3 ) at that distance. Ordinarily an apparatus will not be sensible to the individual photons; the cumulative effect of many photons emitted or absorbed will appear as a continuous, macroscopically observable response.

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  Semiconductor Lasers and Herterojunction LEDs

  • Henry Kressel(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  We note here that (4.4.1) indicates that the normal component of the mag-netic field is continuous across the boundary, and (4.4.2) indicates that the normal component of electric flux density D is discontinuous by an amount equivalent to the surface electric charge density. The tangential field com-ponents are governed by (4.4.3), which indicates that the tangential electric field is continuous across the boundary, and (4.4.4), indicating the tangential magnetic field intensity is discontinuous by an amount equivalent to the surface current density. The condition (4.4.1) results from the fact that there are no magnetic charges, and condition (4.4.2) is derived from Gauss's law. On the other hand, condition (4.4.3) is due to the fact that the electric field is conservative, i.e., the integral of the electric field around a closed path is zero. Condition (4.4.4) is derived from Ampere's circuital theorem. For the case in which the surface charge densities and surface current densi-ties are nonexistent, the boundary conditions reduce to in = B 2n (4.4.5) A . = D la (4.4.6) = E u (4.4.7) (4.4.8) where the subscripts n and t refer to the normal and transverse components, respectively. 4.5 Poynting's Theorem The propagation or flow of electromagnetic energy can be determined from Maxwell's equations. Generally, the discussion of the Poynting theorem is 124 4. RELEVANT CONCEPTS IN ELECTROMAGNETIC FIELD THEORY associated with the conservation of energy: The total energy flowing into a given volume in space is equal to the energy dissipated within the volume. Assuming harmonic time variation Maxwell's equations can be written as H* • V x E = -icoH* • B (4.5.1) E • V X H* — E • J* -f icoE • D* (4.5.2) where H* is the complex conjugate of H. Subtracting (4.5.2) from (4.5.1) and using the vector identity V • (E x H*) = H* - V x E - E .

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