ELECTROMAGNETIC FIELD THEORY

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Maxwell's equations in differential form

Maxwell's equations in differential , or point, form are:

$\displaystyle \nabla \times \underline{\mathcal{H}}$ $\displaystyle =\underline{\mathcal{J}}_e + \frac{\partial \underline{\mathcal{D}}}{\partial t} \space ,$ (1.1)

$\displaystyle \nabla \times \underline{\mathcal{E}}$ $\displaystyle =-\underline{\mathcal{J}}_m - \frac{\partial \underline{\mathcal{B}}}{\partial t} \space ,$ (1.2)

$\displaystyle \nabla \cdot \underline{\mathcal{D}}$ $\displaystyle = \rho_e \space ,$ (1.3)

$\displaystyle \nabla \cdot \underline{\mathcal{B}}$ $\displaystyle = \rho_m \space ,$ (1.4)

where

$\underline{\mathcal{E}}$ = electric field (in V/m) ,

$\underline{\mathcal{H}}$ = magnetic field (in A/m),

$\underline{\mathcal{D}}$ = electric flux density (in ),

$\underline{\mathcal{B}}$ = magnetic flux density (in ),

$\rho_e$ = electric charge density (in ),

$\underline{\mathcal{J}}_e$ = electric current density (in ),

$\rho_m$ = magnetic charge density (in ),

$\underline{\mathcal{J}}_m$ = magnetic charge density (in ),

and all eight quantities are, in general, functions of position $\underline{r}$ and time t. The operator $\nabla$ is the del or ``nabla'' operator, $\nabla \cdot$ is the divergence operator, and $\nabla \times$ is the curl operator.
Presently, there is no experimental evidence to confirm the existence of isolated magnetic charges; therefore, many authors set both $\rho_m$ and $\underline{\mathcal{J}}_m$ equal to zero at the onset, in equations (1.1) and (1.4). There are, however, at least two reasons for retaining $\rho_m$ and $\underline{\mathcal{J}}_m$ : first, symmetry between electric and magnetic quantities is retained in (1.2-1.4); second, equivalent magnetic sources appear in a variety of applications, such as radiation and scattering from aperture antennas or penetrable bodies.
By taking $\nabla \cdot$ of (1.1) and using (1.3) and the identity $\nabla \cdot \nabla \times \equiv 0$ , and by assuming the interchangeability of space and time differentiation, we obtain the continuity equation

$\displaystyle \nabla \cdot \underline{\mathcal{J}}_e + \frac{\partial \rho_e}{\partial t} =0$ (1.5)

which expresses conservation of electric charge . Similarly, from (1.1) and (1.4) we obtain the continuity equation

$\displaystyle \nabla \cdot \underline{\mathcal{ J}}_m + \frac{\partial \rho_m}{\partial t} = 0 \space ,$ (1.6)

which expresses the conservation of magnetic charge .
Conversely, by taking $\nabla \cdot$ of (1.1) and using (1.5), or $\nabla \cdot$ of (1.2) and using (1.6), we find out that the two quantities $(\nabla \cdot \underline{\mathcal{ D}} -\rho_e)$ and $(\nabla \cdot \underline{\mathcal{ B}} - \rho_m)$ are independent of time; hence (1.3) and (1.4) follow, provided that they be valid at a given time (this is the case if, for example, the universe has a finite lifetime). Thus, we may consider (1.1-1.4) as the fundamental equations of electromagnetism, in which case (1.5) and (1.6) are derivative equations, or we may consider the curl equations (1.1) and (1.2) and the continuity equations (1.5) and (1.6) as the fundamentals equations, in in which case the divergence equations (1.3) and (1.4) become derivative equations on the assumption of a finite lifetime for our universe. We shall consider (1.1-1.4) as our fundamental equations.
Equations (1.1-1.4) are necessary but not sufficient for the determination of the eight field quantities (six vectors and two scalars) which appear in them. We still must specify the primary sources of the electromagnetic fields, as well as the physical properties of the medium in which the field exists; these physical properties take the form of functional relations among the various field quantities, called constitutive relations, which will be examined detail in Chapter 2. Meanwhile, we observe that in vacuo (or free) the constitutive relations are:

$\begin{displaymath}\begin{array}{cc} \underline{\mathcal{D}} = \varepsilon_0 \un...
...mathcal{ B}}=\mu_0 \underline{\mathcal{H}} \space , \end{array}\end{displaymath}$ (1.7)

where $\varepsilon_0$ and $\mu_0$ are two constants called the electric permettivity and the magnetic permeability of free space. The values of $\varepsilon_0$ and $\mu_0$ depend on the system of units adopted. We use the rationalized MKSA system in which

$\displaystyle \mu_0= 4\pi {10}^{-7} \frac{\mbox{henry}}{\mbox{meter}} \left( \frac{\mathrm{H}}{\mathrm{m}} \right)$ (1.8)

by definition, while $\varepsilon_0$ is obtained from the formula

$\displaystyle c_0=\frac{1}{\sqrt{\varepsilon_0 \mu_0}}$ (1.9)

where is the velocity of light in free space ( $c_0 \approx 2.997 \, {10}^8 \mathrm{m}/\mathrm{s}$ ); $\varepsilon_0$ is measured in $\mathrm{farad}/\mathrm{meter} \left(\mathrm{F}/\mathrm{m} \right)$ .
Maxwell's equations in free space are obtained by substituting (1.7) into (1.1-1.4):

$\displaystyle \nabla \times \underline{\mathcal{H}}$ $\displaystyle =\underline{\mathcal{J}}_e + \varepsilon_0 \frac{\partial \underline{\mathcal{ E}}}{\partial t} \space ,$ (1.10)

$\displaystyle \nabla \times \underline{\mathcal{E}}$ $\displaystyle =-\underline{\mathcal{J}}_m - \mu_0 \frac{\partial \underline{\mathcal{H}}}{\partial t} \space ,$ (1.11)

$\displaystyle \nabla \cdot \underline{\mathcal{E}}$ $\displaystyle = \frac{\rho_e}{\varepsilon_0} \space ,$ (1.12)

$\displaystyle \nabla \cdot \underline{\mathcal{H}}$ $\displaystyle = \frac{\rho_m}{\mu_0}\space .$ (1.13)

Next: Maxwell's equations in transform Up: Maxwell's equations Previous: Maxwell's equations in differential

Maxwell's equations in integral form
Consider a fixed volume bounded by the closed surface , whose outward unit normal is $\vec{n}$ as shown in Fig. 1.1.

Figure 1.1: Geometry for Gauss theorem.

$\begin{figure}
\begin{center}
\mbox{}
\centerline{\psfig{figure=chap1/Gauss.eps,height=6cm}}
\end{center}\end{figure}$

Integration of (1.3,1.4) and (1.5,1.6) over , followed by the use of the divergence theorem, yields:

$\displaystyle \oint_S \underline{\mathcal{ D}} \cdot \vec{n} \, dS$ $\displaystyle = \int_v \rho_e \, dv = Q_e \space ,$ (1.14)

$\displaystyle \oint_S \underline{\mathcal{ B}} \cdot \vec{n} \, dS$ $\displaystyle = \int_v \rho_m \, dv = Q_m \space ,$ (1.15)

$\displaystyle \oint_S \underline{\mathcal{ J}}_e \cdot \vec{n} \, dS$ $\displaystyle = -\frac{dQ_e}{dt} \space ,$ (1.16)

$\displaystyle \oint_S \underline{\mathcal{ J}}_m \cdot \vec{n} \, dS$ $\displaystyle = - \frac{dQ_m}{dt} \space ,$ (1.17)

where and are the total electric charge (in C) and magnetic charge (in Wb) inside the volume , respectively.
Equation (1.14) means that the total electric charge contained in equals the outgoing flux of $\underline{\mathcal{D}}$ through the surface of . A similar interpretation applies to (1.15); since is zero, the total outgoing flux of $\underline{\mathcal{B}}$ through any fixed closed surface is zero.
Equation (1.16) means that the rate of decrease of the total electric charge inside equals the amount of electric charge which leaves in unit time by traveling outward through ; thus, (1.3) is obviously a statement of conservation of electric charge. A similar interpretation applies to (1.17); since is zero, the outgoing flux of through is also zero.
As a particular application of (1.14), consider a point charge located at the center of a sphere of radius in free space. Because of symmetry $\underline{\mathcal{D}}$ is radially directed and has the same magnitude at all points on . Hence (1.14) gives:

$\displaystyle \underline{\mathcal{ D}}=\frac{Q_e}{4 \pi r^2} \hat{r}$ (1.18)

and, with the use of (1.7):

$\displaystyle \underline{\mathcal{ E}}=\frac{Q_e}{4 \pi \varepsilon_0 r^2} \hat{r}$ (1.19)

which is Coulomb's law of electrostatic.
Let us now consider a fixed open surface bounded by a closed curve , as shown in Fig. 1.2. The unit normal $\hat{n}$ on and the unit tangent $\hat{l}$ along are chosen according to the right-handed corkscrew rule. Integration of (1.1-1.2) over and use of Stokes' theorem yields:

$\displaystyle \oint_l \underline{\mathcal{ H}} \cdot \, d\underline{l}$ $\displaystyle = \int_S \underline{\mathcal{ J}}_e \cdot \hat{n} \, dS + \frac{d}{dt} \int_S \underline{\mathcal{ D}} \cdot \hat{n} \, dS \space ,$ (1.20)

$\displaystyle \oint_l \underline{\mathcal{ E}} \cdot \, d\underline{l}$ $\displaystyle =- \int_S \underline{\mathcal{ J}}_m \cdot \hat{n} \, dS - \frac{d}{dt} \int_S \underline{\mathcal{ B}} \cdot \hat{n} \, dS \space ,$ (1.21)

Figure 1.2: Geometry for Stokes theorem.

$\begin{figure}
\begin{center}
\mbox{}
\centerline{\psfig{figure=chap1/Stokes.eps,height=6cm}}
\end{center}\end{figure}$

For time-invariant, or stationary, fields, eq. (1.20) becomes:

$\displaystyle \oint_l \underline{\mathcal{ H}} \cdot \, d\underline{l}=\int_S \underline{\mathcal{ J}}_e \cdot \hat{n} \, dS=I_e \space ,$ (1.22)

where is the total electric current which flows through the closed loop . Equation (1.22) is Ampère's law, whose generalization to time-varying fields requires the addition of the second term in the right-hand side of (1.20); this added term, whose existence was postulated by Maxwell in 1861, is called the displacement current.
For the case $\underline{\mathcal{ J}}_m \equiv 0$ , equation (1.21) is Lenz's law which in turn represents a generalization of Kirchhoff's second law of circuit theory to the case of time-varying fields by the inclusion of an induction term.

Next: Maxwell's equations in transform Up: Maxwell's equations Previous: Maxwell's equations in differential

1999-07-01

Next: Maxwell's equations in integral Up: Maxwell's equations Previous: Maxwell's equations

1999-07-01