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.