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.

20. A Qualitative Overview of MHD: From Electrodynamics to MHD

The fluidity of the conductor, which makes the interaction between u and B difficult to quantify, is the only difference between MHD and conventional electrodynamics. However, many of the important features of MHD are concealed in electrodynamics, and these can be shown by simple experiments.

We are going to introduce some notation. Let μ be the permeability of free space, σ is the electrical conductivity, ρ is the density of the conducting medium and l is the characteristic length scale. There are three important parameters in MHD:

Magnetic Reynolds number: ${{R}_{m}}=\mu \sigma ul$                                  (20.1)

Alfvèn velocity: ${{v}_{a}}={B}/{\sqrt{\rho \mu }}\;$                                       (20.2)

Magnetic damping time: $\tau ={{\left( {\sigma {{B}^{2}}}/{\rho }\; \right)}^{-1}}$                                      (20.3)

The first parameter (20.1) is dimensionless measure of conductivity, the second parameter (20.2) has the dimension of speed and the third parameter (20.3) has the dimension of time as their names suggest.

Magnetic Reynolds number is one of the important dimensionless parameters in MHD. In case where R_m is large, the magnetic field lines tend to be ‘frozen’ together with the conducting medium. As consequences, firstly, during fluid’s motion, magnetic flux passing through any closed material loop tends to be conserved and secondly, small disturbances tend to result in near-elastic oscillation with the magnetic field providing the restoring force for the vibration. This results in Alfvèn waves with frequency of $\omega \sim {{v}_{a}}/l$.

When R_m is small, the fluid’s velocity u will has small influence on the magnetic field B and the induced field is very small by comparison with the imposed field. We shall see that the magnetic field behaves quite distinctively, it is dissipative in nature rather than elastic, mechanical motion is damped by converting kinetic energy into heat via Joule dissipation. In this case, the relevant time scale is the damping time τ rather than $l/{{v}_{a}}$.

Let us now start with the elementary laws of electromagnetism, i.e. Ohm’s law, Faraday’s law and Ampere’s law.  

The Ohm’s law (Figure 20.1) for stationary conductors says empirically that: $\mathbf{J}=\sigma \mathbf{E}$ , where E is the electric field and J is the density of the current. Analogically, this law can be interpreted as J being proportional to the Coulomb force $\mathbf{f}=q\mathbf{E}$ which acts on the  free charge carriers, with q being their charge. However, the free charges will experience an extra force $q\mathbf{u}\times \mathbf{B}$ when the conductor moves in a magnetic field with velocity u, and the Ohm’s law becomes:

$\mathbf{J}=\sigma (\mathbf{E}+\mathbf{u}\times \mathbf{B})$                                                 (20.4)

The quantity $\mathbf{E}+\mathbf{u}\times \mathbf{B}$ is also called the effective electric field E_r, which is the total electromagnetic force per unit charge, and is measured in a frame of reference moving with velocity u relative to the laboratory frame.

${{\mathbf{E}}_{\mathbf{r}}}=\mathbf{E}+\mathbf{u}\times \mathbf{B}=\mathbf{f}/q$                                              (20.5)

Figure 20.1. illustrates Ohm’s law in (a) stationary conductors and (b) moving conductors.

The Faraday’s law is given by:

$emf=\oint\limits_{c}{{{\mathbf{E}}_{r}}\cdot d\mathbf{l}}=-\frac{d}{dt}\int\limits_{S}{\mathbf{B}\cdot d\mathbf{S}}$                                        (20.6)

Here, C is a closed curve composed of line elements dl. The curve might be fixed in space or move with the conducting medium. S is any surface which spans C.

Figure 20.2. illustrates the Faraday’s law which describes the e.m.f  generated by: (a) the movement of a conductor, and (b) by a time-dependent magnetic field.

This law tells about the e.m.f. which is generated in a conductor as a result of a time-dependent magnetic field due to the motion of a conductor within a magnetic field.

Then, there’s Ampere’s law that tells us about the magnetic field associated with a given distribution of current J. If C is a closed curve and S is any surface spans that curve, the Ampere’s law states that:

$\oint\limits_{C}{\mathbf{B}\cdot d\mathbf{l}=\mu \int\limits_{S}{\mathbf{J}\cdot d\mathbf{S}}}$                                                     (20.7)

Figure 20.3 Ampere’s law applied to a wire.

Finally, the force F per unit volume of the conductor is given by:

$\mathbf{F}=\mathbf{J}\times \mathbf{B}$                                                 (20.8)

Equation (20.8) is the Lorentz force, which can be derived from the force acting on individual charge carriers $\mathbf{f}=q(\mathbf{u}\times \mathbf{B})$.