21. MHD: Some Important Equations of Electrodynamics (1)

In most of our cases, we are concerned with non-magnetic and conducting materials. We assume that the material properties, such as conductivity, σ, are spatially uniform and incompressible. In this chapter, we are going to discuss about the electric force and the Lorentz force.

Generally, a particle moving with velocity u and carrying electric charge q, is subjected to electromagnetic forces and governed by the following equation:

$\mathbf{f}=q{{\mathbf{E}}_{s}}+q{{\mathbf{E}}_{i}}+q\mathbf{u}\times \mathbf{B}$                                              (21.1)

Here, E_s is the electrostatic field and the first term is the electrostatic force or Coulomb force which arises from the mutual attraction or repulsion of electric charges. The second term is the force which arises in the presence of time-varying magnetic field, since E_i is the electric field induced by the varying magnetic field. Due to the motion of the charge in a magnetic field, Lorentz force arises, and this is given by the third term $q\mathbf{u}\times \mathbf{B}$.

Coulomb’s law says that electrostatic field E_s is irrotational, and Gauss’s law fixes the divergence of E_s. These laws are respectively given by:

$\nabla \cdot {{\mathbf{E}}_{s}}={}^{{{\rho }_{e}}}/{}_{{{\varepsilon }_{0}}}$                                                     (21.2)

$\nabla \times {{\mathbf{E}}_{s}}=0$                                                        (21.3)

ρ_e is the total charge density (free plus bound charges) and ε₀ is permittivity of free space. Introducing electrostatic potential V, defined by ${{\mathbf{E}}_{s}}=-\nabla V$, equation (21.2) becomes:

${{\nabla }^{2}}V=-{}^{{{\rho }_{e}}}/{}_{{{\varepsilon }_{0}}}$                                                   (21.4)

On the other hand, induced electric field E_i has zero divergence, and finite rotational component. This is governed by Faraday’s law:

$\nabla \cdot {{\mathbf{E}}_{i}}=0$                                                      (21.5)

$\nabla \times {{\mathbf{E}}_{i}}=-\frac{\partial \mathbf{B}}{\partial t}$                                                      (21.6)

Conveniently, we define the total electric field as: E = E_s + E_i and so we have:

Gauss’s law: $\nabla \cdot \mathbf{E}={}^{{{\rho }_{e}}}/{}_{{{\varepsilon }_{0}}}$                                            (21.7)

Faraday’s law: $\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$                                              (21.8)

Electrostatic and Lorentz force: $\mathbf{f}=q(\mathbf{E}+\mathbf{u}\times \mathbf{B})$                           (21.9)

Equations (21.7) and (21.8), i.e. Gauss’s and Faraday’s law, determine uniquely the electric field since the divergence and curl of the field are known and the boundary conditions are specified. Equation (21.9) defines the electric field E and magnetic field B, where E is the force per unit charge on a test charge at rest in the observer’s frame. If the charge is moving, the additional force $q\mathbf{u}\times \mathbf{B}$ appears which is used to define B. However, Newtonian relativity which is all that is required for MHD tells us that the electric force relative to a moving frame f_r is equal to the electric force due to electric field at rest f, i.e. f_r = f.

7.Plane Waves

One of the simple examples of a three-dimensional wave is the plane wave. In order to derive the mathematical expression of a plane wave moving in direction vector $\vec{k}$ perpendicular to the plane, we first write the position vector in Cartesian coordinates in terms of unit vectors $\hat{\vec{i}},\hat{\vec{j}},\hat{\vec{k}}$  (see Figure 7.1.a):

$\vec{r}=x\hat{\vec{i}}+y\hat{\vec{j}}+z\hat{\vec{k}}$                                                          (7.1)

For a plane anywhere in space, that begins at some arbitrary origin O and ends at the point (x,y,z) on that plane (see Figure 7.1.b):

$(\vec{r}-{{\vec{r}}_{o}})=(x-{{x}_{o}})\hat{\vec{i}}+(y-{{y}_{o}})\hat{\vec{j}}+(z-{{z}_{o}})\hat{\vec{k}}$                                        (7.2)

Figure 7.1. (a) Cartesian unit vectors. (b) A plane wave moving in the $\vec{k}$-direction.

We would set the vector $\left( \vec{r}-{{{\vec{r}}}_{o}} \right)$ to sweep out a plane perpendicular to $\vec{k}$ therefore,

$\left( \vec{r}-{{{\vec{r}}}_{o}} \right)\bullet \vec{k}=0$                                                              (7.3)

Rewriting directional vector $\vec{k}$ in term of unit vectors:

$\vec{k}={{k}_{x}}\hat{\vec{i}}+{{k}_{y}}\hat{\vec{j}}+{{k}_{z}}\hat{\vec{k}}$                                                       (7.4)

And Equation (7.3) becomes:

${{k}_{x}}\left( x-{{x}_{o}} \right)+{{k}_{y}}\left( y-{{y}_{o}} \right)+{{k}_{z}}\left( z-{{z}_{o}} \right)=0$                                       (7.5)

Thus, $\vec{k}\bullet \vec{r}=$ constant.

A set of planes over which the wave function $\psi (\vec{r})$ varies sinusoidally for constant $\vec{k}\bullet \vec{r}$  can be constructed in term of complex representation:

$\psi (\vec{r})=A{{e}^{i\vec{k}\bullet \vec{r}}}$                                                         (7.6)

Naturally, these harmonic functions are repetitive, therefore, they can be expressed by:

$\psi (\vec{r})=\psi (\vec{r}+\frac{\lambda \vec{k}}{k})$                                                    (7.7)

In this equation, k is the magnitude of $\vec{k}$ and ${}^{{\vec{k}}}/{}_{k}$ is a unit vector parallel to it (see Figure 7.2).

Figure 7.2. Plane waves.

Applying Equation (7.7) to Equation (7.6), we obtain the exponential form of the harmonic wave function:

$\psi (\vec{r})=A{{e}^{i\vec{k}\bullet \vec{r}}}=A{{e}^{i\vec{k}\bullet (\vec{r}+{}^{\lambda \vec{k}}/{}_{k})}}=A{{e}^{i\vec{k}\bullet \vec{r}}}{{e}^{i\lambda k}}$                                         (7.8)

To satisfy Equation (7.6), we must have:

${{e}^{i\lambda k}}=1={{e}^{i2\pi }}$                                                           (7.9)

Therefore $\lambda k=2\pi $ or $k={}^{2\pi }/{}_{\lambda }$. The vector $\vec{k}$ is called the propagation vector, whose magnitude is the propagation number k.

Since $\psi (\vec{r})$ is moving i.e. is varying in time, we introduce the time dependence ωt in analogy to that of the one-dimensional wave equation:

$\psi \left( \vec{r},t \right)=A{{e}^{i(\vec{k}\bullet \vec{r}\mp \omega t)}}$                                                    (7.10)

With A, ω, and k constant. The wave travels in the direction of propagation vector $\vec{k}$ such that we can assign a phase corresponding to it at each point in space and time. The surfaces connecting every points of equal phase at any given time are known as the wavefronts (see Figure 7.3). 

Figure 7.3. Wavefronts for a harmonic plane waves.

The propagation velocity of the wavefront is equal to the phase velocity of a plane wave given by Equation (7.10). Following from Figure 7.2, r_k is the scalar component of $\vec{r}$ in the direction of $\vec{k}$. Since we assume that the wave is homogeneous and the disturbance on a wavefront is constant, so that after a time dt and after the wavefront moves along $\vec{k}$ by a distance dr_k, Equation (7.10) should be:

$\psi (\vec{r},t)=\psi (r+d{{r}_{k}},t+dt)=\psi ({{r}_{k}},t)$                                             (7.11)

Which can be rewritten in exponential form as follows:

$A{{e}^{i(\vec{k}\bullet \vec{r}\mp \omega t)}}=A{{e}^{i(k{{r}_{k}}+kd{{r}_{k}}\mp \omega t\mp \omega dt)}}=A{{e}^{i(k{{r}_{k}}\mp \omega t)}}$                                      (7.12)

So that $kd{{r}_{k}}=\pm \omega dt$ and the wave velocity ${}^{d{{r}_{k}}}/{}_{dt}$ is then:

$\frac{d{{r}_{k}}}{dt}=\pm \frac{\omega }{k}=\pm v$                                                         (7.13)