Electromagnetism

Abali, Bilen Emek

doi:10.1007/978-981-10-2444-3_3

Bilen Emek Abali⁵

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 55))

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Abstract

Electromagnetism is the theory of electromagnetic interactions with matter. In this theory there occur various new quantities; and this makes a straight-forward introduction of equations challenging. These new quantities lead to electromagnetic fields, which can be measured. However, they often lack a clear interpretation. For example, a moving electric charge fails to be understood completely, but an electric current is something we use in our daily lives. In order to establish a knowledge of electromagnetism, we will motivate governing equations one-by-one with applications. We consider a conducting wire and investigate how it heats up due to the production term in the balance of internal energy. We introduce electric field and magnetic flux in polarized materials. By using thermodynamical principles we derive the constitutive equations and solve a problem addressing the thermoelectric coupling. Then we include plasticity. Moreover, a piezoelectric sensor is discussed by deriving the constitutive equations in a thermodynamically consistent way. A magnetohydrodynamical problem is presented in the last section. All applications are discussed theoretically and implemented by using open-source codes.

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Notes

1.
They are named after James Clerk Maxwell.
2.
See [30].
3.
It is named for André Marie Ampère and Hendrik Antoon Lorentz.
4.
The abbreviation mks stands for meter, kilogramm, seconds.
5.
It is named after Giovanni Giorgi.
6.
It is named for Charles-Augustin de Coulomb.
7.
The electromagnetic theory started by believing that electric field $E_i$ and magnetic field $H_i$ shall be the primitive variables. In 1940s the theoretical thermodynamics introduced the way of starting with $E_i$ and $B_i$ fields. There are still explanations starting different than the approach used herein.
8.
The units are named after Alessandro Volta and Nikola Tesla.
9.
In the case of an open system, there would be an additional convective term.
10.
Charge per time, C/s, is called A(mpere) named for André Marie Ampère. The electric current (area) density, $J_i$, is in A/m$^2$.
11.
We know that the action fails to be instantaneous; however, our experimental machine is detecting slower than the information transported from one end to the other end of the wire.
12.
Since the continuum body is defined as a collection of charges, density here means a quantity per electric charge, hence we write a force charge density meaning a force per charge, N/C. Another adequate name is intensity, which we shall use further on.
13.
The electromotive intensity is measured in V(olt) per meter since 1 V is the energy per electric charge, 1 V $\hat{=}$ 1 J/C such that 1 N/C $\hat{=}$ 1 N m/(C m) $\hat{=}$ 1 V/m. Moreover, a single V across a wire conducting a current of 1 A dissipates of 1 W $\hat{=}$ 1 J/s of power, hence 1 W $\hat{=}$ 1 A V.
14.
The unit of magnetic flux area density is in T or in Wb (Weber) per m$^2$ named after Wilhelm Eduard Weber. Since 1 Wb $\hat{=}$ 1 kg m$^2$/(C s) $\hat{=}$ 1 N s m/C $\hat{=}$ 1 V s, magnetic flux area density is in V s/m$^2$.
15.
The force is named for Hendrik Antoon Lorentz. For a detailed physical interpretation of this force in the atomic scale, see [33, Sect. 4–5].
16.
It is named after Georg Simon Ohm.
17.
Electrical conductivity is in S(iemens)/m where S $\hat{=}$ 1/$\Omega $ $\hat{=}$ A/V. Thus the conductivity is in A/(V m). The unit S is named for Werner von Siemens. The unit $\Omega $ is capital omega in Greek pronounced as Ohm if used as the unit named after Georg Simon Ohm.
18.
We refer to [24, Sect. 9] for the transformation of $\mathscr {E}_i$ and $\mathscr {B}_i$ fields from the co-moving to the laboratory frame.
19.
Mathematically speaking the function has to belong to $C^2$ or higher in order to ensure the twice differentiability condition. Since $\phi $ and $\updelta \phi $ are of the same space according to the Galerkin type finite element method, both have to be twice differentiable.
20.
In this section we deal with unpolarized systems.
21.
We should model a rounded wire since in reality wires have round cross sections, however, it is simpler and shorter to code it this way. The presented solutions and discussions are the same for both cases.
22.
A material surface is a material system without convection terms where the domain is a surface instead of a volume leading to an area density instead of a volume density.
23.
It is named after Michael Faraday.
24.
A relation holds universally, if it is free of any dependence on the underlying material. In other words, a universal relation holds for all materials and even in the case of no material—vacuum.
25.
The German word ansatz has the equal meaning of a trial function. We simply find out by trial the functions satisfying differential equations.
26.
For the motivation of the gauge freedom see Appendix A.6 on p. 305.
27.
This gauge is named after Carl Friedrich Gauß.
28.
The gauge is named for Ludvig Valentin Lorenz.
29.
The line element is directed along the positive surface boundary. The positive direction is such that we “walk along” the surface boundary and the surface is on our left-hand side.
30.
They are named after James Clerk Maxwell and Hendrik Antoon Lorentz.
31.
Free space is a technical definition used as a reference for electromagnetic fields, $E_i$, $B_i$. It can be visualized as a perfect vacuum without any medium such as massive particles that may transport the electromagnetic fields. Even in this free space the fields do propagate (with the speed of light, c).
32.
Universal constants hold for every material, even without matter (in free space).
33.
Electronic configurations: Copper (Cu) $3d^{10} 4s^1$, Silver (Ag) $4d^{10} 5s^1$, Gold (Au) $4f^{14} 5d^{10} 6s^1$.
34.
Electronic configuration: Aluminum (Al) $3s^2 3p^1$.
35.
Electronic configurations: Iron (Fe) $3d^6 4s^2$, Titanium (Ti) $3d^2 4s^2$.
36.
This model fails to be correct since if electrons would rotate they would radiate electromagnetic waves. Since experimentally we cannot detect any radiation from atoms this visualization is false. Better models are proposed by using quantum mechanics. However, we keep up with continuum mechanics; for introducing magnetic polarization we use the nice visualization of Rutherford–Bohr’s model named for Ernest Rutherford and Niels Henrik David Bohr.
37.
The magnetization used for the modeling, $\mathscr {M}_i$, is an objective quantity.
38.
The permittivity is measured in F(arad)/m $\hat{=}$ C/(V m) $\hat{=}$ A s/(V m) where F is named after Michael Faraday. The permeability is measured in H(enry)/m $\hat{=}$ Wb/(A m) $\hat{=}$ V s/(A m) where H is named for Joseph Henry.
39.
There are various methods for measuring the susceptibilities, see for example [26, 37].
40.
The interface is a fictitious surface without mass. If we have a thin layer between two different materials, we may declare it as a singular surface (surface has zero thickness) by neglecting the layers thickness. However, the singular would have then a mass. We consider herein singular surfaces without mass.
41.
In many applications the surface charges have no effect at all. In Sect. 3.5 on p. 243 we will simulate the piezoelectric effect under 100 V and have a small error less than 1 V by neglecting the surface charges, see for a detailed computation of surface charges in piezoelectric ceramics in [21]. For some applications concerning mass diffusion (electromigration) in mixtures, the surface charges may have a significant effect. In this book mixtures are out of scope.
42.
The interchangeability of the order of variables in a differentiation is named after Hermann Amandus Schwarz.
43.
PTFE stands for PolyTetraFluoroEthylene—its prominent brand-name is Teflon from DuPont in France.
44.
See Appendix A.3 on p. 297 for instructions how to mark the surfaces for applying the boundary conditions and to mark the volumes for different parts.
45.
In reality, the transformer is housed in a polymer like epoxy, which is an insulator alike air. We just neglect the electric polarization occurring in the polymer housing.
46.
Eddy current was discovered firstly by Jean Bernard Léon Foucault.
47.
Der Litzendraht in German means stranded wire.
48.
For a scalar and as a special case for the velocity the total time rate, $\frac{\mathrm d(\cdot )}{\mathrm dt}$, is equal to the objective time rate, , for a fixed coordinate system, $w_i=0$.
49.
From a theoretical point of view this assumption is not satisfying. We shall consider $T=0$ state as the zero state for entropy. At $T=T_{\text {ref}}$ the entropy is then $\eta _0$ and it is unknown. Since we only employ the rate of entropy, the unknown value drops and in the end we reach the same formulation as presented herein. However, for strains the coefficient of thermal expansion, $\alpha _{ij}$, has been measured by using a reference temperature, which is certainly not 0 K. In simulations we use $T_{\text {ref}}=300$ K.
50.
In the literature the Onsager relation is motivated by microscopic calculations. Herein we reach the same conclusion by using thermodynamics. Onsager’s relations are named after Lars Onsager.
51.
It is called for William Thomson (Lord Kelvin).
52.
It is named after Thomas Johann Seebeck.
53.
This effect is named after Jean Charles Athanase Peltier.
54.
The material never reaches an ultimate strain, where a failure is expected. The accumulated deformation causes such a failure over time. For an electronic device this time frame is more than couple of years.
55.
We have introduced associated plasticity in Sect. 1.6 and applied into the thermodynamical formulation in Sect. 3.3.
56.
Measurement of the stiffness tensor is undertaken in Fraunhofer IZM in Berlin (Germany) for a typical laminate used in many circuit boards, the values at the room temperature are taken from [3], the values between 233–443 K can be found in [2, Table 1].
57.
The factor 2/3 is due to the tensile test. See Sect. 1.6 for a motivation of the deviatoric plastic strain.
58.
Pierre Curie had been the husband of Marie Skłodowska-Curie.
59.
PZT is the general name for lead-zirconate-titanate, Pb$_{1.1}$Zr$_{0.3}$Ti$_{0.7}$O$_3$ and PbZr$_{0.52}$Ti$_{0.48}$O$_3$ are the common used compositions of PZT.
60.
See [15, Chap. XIV].
61.
The same formulation in Eq. (3.190) can also be found by using an electromagnetic energy-momentum tensor, $S_{ij}$, where this tensor is just the negative of the electromagnetic stress, $S_{ij}=-m_{ij}$. Confusingly, in some books the divergence is taken regarding the second index. For the electromagnetic stress as well as for the energy-momentum tensor the choice matters, since they are not required to be symmetric.
62.
See for example [14, 25, 29].
63.
The different electromagnetic momenta are named after John Henry Poynting, Hermann Minkowski, Max Abraham.
64.
Thought experiments.
65.
See [8] for a review about experimental evidences for each choice. In [7, Sect. 4.4 and 8.7] there is some insight about different choices of electrodynamic forces and energies in the literature. See [20] for an experimental discussion about Abraham and Minkowski choices. See [6] for simulations of experiments with different choices and their comparison to experiments from the literature.
66.
It is named for James Clerk Maxwell.
67.
A thermodynamical formulation for the choice of $\mathscr {G}_i^\text {A}$ can be found in [15, Chap. XIV, Sect. 2].
68.
See [15, Chap. XIV, Sect. 2].
69.
An alternative is to use the following formulation:
$$\begin{aligned} \begin{gathered} \rho _0 \frac{\mathrm du}{\mathrm dt} = \rho _0 T \frac{\mathrm d\eta }{\mathrm dt} + ( \,^\text {r}\!\sigma _{ji} - \,^\text {r}\!P_j E_i + \,^\text {r}\!\mathscr {M}_i B_j )\frac{\mathrm d\varepsilon _{ij}}{\mathrm dt} - \,^\text {r}\!P_i \frac{\mathrm dE_i}{\mathrm dt} + B_i \frac{\mathrm d\,^\text {r}\!\mathscr {M}_i}{\mathrm dt} \ . \end{gathered} \end{aligned}$$
This formulation leads to the same constitutive equations and is also admissible. The argumentation for explaining the measurements in the dual variables becomes more difficult by using the alternative formulation.
70.
See the measurements described in [27].
71.
It is named after Edwin Herbert Hall.
72.
Some materials have different stable states where the configuration of molecules are different. By applying a high electric field, i.e., poling, the configuration is changed to the new stable state. The lattice is distorted in this new state so that the electric field and deformation are connected. An electric field (smaller than the poling field) induces a strain (distortion in the lattice). This shape change is reversible as long as the applied field remains smaller than the poling field.
73.
All parameters of PZT-5H are compiled from http://www.piceramic.com/piezo-technology/fundamentals.html and http://bostonpiezooptics.com/ceramic-materials-pzt
74.
Iron with carbon is named as steel.
75.
In 1886 Charles Martin Hall and (his sister) Julia Brainerd Hall developed the process. In the same year Paul Héroult did develope the same process, independently to them. Therefore, the process is called after all of them.
76.
There are many different ores containing aluminum. Mainly bauxite is used. It is a mixture of aluminum minerals, clay minerals, and insoluble materials. The main aluminum minerals in bauxite are gibbsite Al(OH)$_3$, boehmite $\gamma -$AlO(OH), and diaspore $\alpha -$AlO(OH).
77.
There are many investigations for optimizing the extraction conditions, for example, see [5, 36].
78.
The chemical composition of cyrolite is Na$_3$AlF$_6$.
79.
In solid bodies the so-called equation of state (EOS) is well-documented for many materials. For example, Mie-Gruneisen approximation, named after Gustav Adolf Feodor Wilhelm Ludwig Mie and Eduard Grüneisen, is used for a relation between energy and pressure, in our notation $\partial {\mathscr {g}}/ \partial \bar{p} = v /\Gamma $, where $\Gamma $ denotes the Gruneisen parameter. See for values of such a parameter for Cu in [19] or for Cu, Al, Pb in [17, Table 1].
80.
See [23, Fig. 5].
81.
For a copper cell simulation see [34] and for an aluminum cell simulation see [13]. A review of such simulations can be found in [39].
82.
See [32, 35].
83.
See [16].
84.
For example in [9–11, 38].

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Bilen Emek Abali

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Abali, B.E. (2017). Electromagnetism. In: Computational Reality. Advanced Structured Materials, vol 55. Springer, Singapore. https://doi.org/10.1007/978-981-10-2444-3_3

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DOI: https://doi.org/10.1007/978-981-10-2444-3_3
Published: 23 October 2016
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