3.11: Magnetic Korteweg-Helmholtz Force Density

Thermodynamic techniques for determining the magnetization force density are analogous to those outlined for the polarization force density in Sec. 3.7. In fact, if there were no free current density, the magnetic field intensity, like the electric field intensity, would be irrotational. It would then be possible to make a derivation that would be the complete analog of that for the polarization farce density. However, in the following the force density due to free currents is included and hence \(\overrightarrow{H}\) is not irrotational.

The constitutive law takes the form

\[ \overrightarrow{H} = \overrightarrow{H} (\alpha_1, \alpha_2…\alpha_m, \overrightarrow{B}) \, or \, \overrightarrow{B} = \overrightarrow{B} (\alpha_1, \alpha_2…\alpha_m, \overrightarrow{H}) \label{1} \]

with specific possibilities given in Table 3.7.1 with \(\varepsilon \rightarrow \mu, \, \overrightarrow{e} \rightarrow \overrightarrow{H} \) and \(\overrightarrow{D} \rightarrow \overrightarrow{B}\). A conservative electro-mechanical subsystem is assembled mechanically, with no electrical excitations, so that it assumes a configuration identical to the one for which the force density is required. By the 'definition of the subsystem, this process requires no energy. Then, with the mechanical system fixed (the a's fixed),electrical excitations are applied so as to establish the free currents in excitation coils and in the medium itself, with the distribution that for which the force density is required. This procedure is formalized in Sec. 2.12 and a system schematic is shown in Fig. 2.14.2. As was shown in Sec. 2.14, currents in excitation coils are conveniently regarded as part of the total distribution of free current density. Hence, the volume of interest now includes all of the region permeated by the magnetic field.

Now, with the electrical excitations established, a statement of conservation of energy, with the electrical excitations held fixed but the material undergoing an incremental displacement, is Eq. 3.7.7, where now \(W\) is the magnetic energy density given from Eq. 2.14.10 by

\[ W = \int_o^{\overrightarrow{B}} \overrightarrow{H} ( \alpha_1, \alpha_2…\alpha_m, \overrightarrow{B}^{‘}) \cdot \delta \overrightarrow{B}^{‘} \label{2} \]

The following steps, leading to a deduction of the force density, are analogous to those taken in Sec. 3.7. The link between the \(\alpha\)'s and \(\delta \overrightarrow{\xi}\) is given by Eq. 3.7.5. What is the connection between \(\overrightarrow{J}_f\) and \(\delta \overrightarrow{\xi}\) ?

Actually, it is a link between the flux linkage and \(\overrightarrow{\xi}\) that is appropriate. If the medium is to both support a free current density and be conservative, the material must be idealized as having an infinite conductivity. This means that any open material surface \(S\) (surface of fixed identity) must link a constant flux:

\[ \delta \int_S \overrightarrow{B} \cdot \overrightarrow{n} da = 0 \label{3} \]

One way to make this deduction is to use the integral form of Faraday's law for a contour \(C\) enclosing a surface \(S\) of fixed identity, Eq. 2.7.3b, with \(\overrightarrow{v} = \overrightarrow{v}_s\). Because the medium is perfectly conducting,\(\overrightarrow{E}^{'} = 0\) and what remains of Faraday's law is Equation \ref{3}. From the generalized Leibnitz rule, Eq. 2.6.4, Equation \ref{3} and the solenoidal nature of \(\overrightarrow{B}\) require that

\[ \int_S \delta \overrightarrow{B} \cdot \overrightarrow{n} da + \oint_C (\overrightarrow{B} \times \delta \overrightarrow{\xi}) \cdot \overrightarrow{d} l = 0 \label{4} \]

Stokes's theorem, Eq. 2.6.3, converts the contour integral to a surface integral. Because this surface is arbitrary, the sum of the integrands must vanish. If it is further recognized that \(\delta \overrightarrow{B} = \nabla \times \delta \overrightarrow{A}\), then it follows that

\[ \delta \overrightarrow{A} = \delta \overrightarrow{\xi} \times \overrightarrow{B} \label{5} \]

Thus, there is established the link between material deformations and the alterations of the field that are required if the deformations are to be flux-conserving.

The change in \(W\) associated with the material deformation, called for in the conservation of energy equation, Eq. 3.7.7, is in general

\[ \delta W = \sum_{i=1}^{n}\frac{\partial{W}}{\partial{\alpha_i}} \delta\alpha_i + \frac{\partial{W}}{\partial{\overrightarrow{B}}} \cdot \delta \overrightarrow{B} \label{6} \]

Where, in view of Equation \ref{2},

\[ \frac{\partial{W}}{\partial{B_j}} = H_j \label{7} \]

It is the integral over the total volume \(V\) of \(\delta W\) that is of interest. The integral of the last term in Equation \ref{6} is

\[ \int_V \frac{\partial{W}}{\partial{\overrightarrow{B}}} \cdot \delta \overrightarrow{B} dV = \int_V \overrightarrow{H} \cdot \delta \overrightarrow{B} dV = \int_V \overrightarrow{H} \cdot \nabla \times \delta \overrightarrow{A} dV \label{8} \]

Because the fields decay to zero sufficiently rapidly at infinity that the surface integral vanishes and because Ampere's law, Eq. 2.3.23b, gives \(\nabla \times \overrightarrow{H} = \overrightarrow{J}_f\), integration of the last term in Equation \ref{8} by parts gives

\[ \int_V \frac{\partial{W}}{\partial{\overrightarrow{B}}} \cdot \delta \overrightarrow{B} dV = \int_V \nabla \cdot \delta \overrightarrow{B} dV = \int_V \nabla \cdot (\delta \overrightarrow{A} \times \overrightarrow{H}) dV + \int_V \delta \overrightarrow{A} \cdot \nabla \times \overrightarrow{H} dV = \oint_S \delta \overrightarrow{A} \times \overrightarrow{H} \cdot \overrightarrow{n} da + \int_V \overrightarrow{J} \cdot \delta \overrightarrow{A} dV = \int_V \delta \overrightarrow{A} \cdot \overrightarrow{J}_f dV \label{9} \]

Substitution for \(\delta \overrightarrow{A}\). from Equation \ref{5} finally gives an expression explicitly showing the \(\overrightarrow{\xi} \) dependence:

\[ \int_V \frac{\partial{W}}{\partial{\overrightarrow{B}}} \cdot \delta \overrightarrow{B} dV = \int_V \delta \overrightarrow{\xi} \times \overrightarrow{B} \cdot \overrightarrow{J}_f dV = -\int_V \overrightarrow{J}_f \times \overrightarrow{B} \cdot \delta \overrightarrow{\xi} dV \label{10} \]

Finally, the energy conservation statement, Eq. 3.7.7, is written with \(\delta W\) given by Equation \ref{6} and in turn,\(\delta \alpha_i\) given by Eq. 3.7.5 and the last term given by Equation \ref{10}:

\[ \int_V [ - \sum_{i=1}^{n} \frac{\partial{W}}{\partial{\alpha_i}} \nabla \cdot (\delta \overrightarrow{\xi} \alpha_i) - \overrightarrow{J}_f \times \overrightarrow{B} \cdot \delta \overrightarrow{\xi} + \overrightarrow{F} \cdot \delta \overrightarrow{\xi} ] dV = 0 \label{11} \]

With the objective of writing the first term as a dot product with \(\delta \overrightarrow{\xi}\), the first term is integrated by parts (exactly as in going from Eq. 3.7.13 to Eq. 3.7.14) to obtain

\[ \int_V [ \sum_{i=1}^{n} \alpha_i \nabla \frac{\partial{W}}{\partial{\alpha_i}} - \overrightarrow{J}_f \times \overrightarrow{B} + \overrightarrow{F} ] \cdot \delta \overrightarrow{\xi} dV = 0 \label{12} \]

The integrand must be zero, not because the volume is arbitrary (it includes all of the system involved in the electromechanics) but rather because the virtual displacements \(\delta \overrightarrow{\xi}\) are arbitrary in their distribution. Hence, the force density is

\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} - \sum_{i=1}^{n} \alpha_i \nabla \frac{\partial{W}}{\partial{\alpha_i}} \label{13} \]

The special cases considered in Sec. 3.7 have analogs that similarly follow from Equation \ref{13}. Because what is involved in deriving these forms involves the magnetization term in Equation \ref{13}, and not the free current force density, these expressions can be written down by direct analogy.

Incompressible Media: The convenient form emphasizing the importance of regions where there are property gradients is

\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} + \sum_{i=1}^{n} \frac{\partial{W}}{\partial{\alpha_i}} \nabla \alpha_i \label{14} \]

Incompressible and Electrically Linear: With a constitutive law

\[ \overrightarrow{B} = \mu_o (1 + \chi_m) \overrightarrow{H} = \mu \overrightarrow{H} \label{15} \]

The force density of Equation \ref{13} reduces to

\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} - \frac{1}{2} H^2 \nabla \mu \label{16} \]

Electrically Linear with Magnetization Dependent on Mass Density Alone: With the constitutive law in the form of Equation \ref{15}, but \(\chi_m = \chi_m (\rho) \), where \(\rho\) is the mass density, the force density is the sum of Equation \ref{14} and a magnetostrictive force density taking the form of the gradient of a pressure:

\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} - \frac{1}{2} H^2 \nabla \mu + \nabla (\frac{1}{2} \rho \frac{\partial{\mu}}{\partial{\rho}} H^2) \label{17} \]

Relation to Kelvin Force Density: With the stipulation that \(W = W(\alpha_1, \alpha_2…\alpha_m, \overrightarrow{B}) is a state function, Equation \ref{13} becomes the sum of a Lorentz force density due to the free current density, the Kelvin force density and the gradient of a pressure:

\[ \overrightarrow{F} = \overrightarrow{J}_f \times \mu_o \overrightarrow{H} + \mu_o \overrightarrow{M} \cdot \nabla \overrightarrow{H} + \nabla [ \frac{1}{2} \mu_o \overrightarrow{H} \overrightarrow{H} + W - \overrightarrow{H} \cdot \overrightarrow{B} - \sum_{i=1}^{m} \alpha_i \frac{\partial{W}}{\partial{\alpha_i}} ] \label{18} \]

The discussion of Sec. 3.7 is as appropriate for understanding these various forms of the magnetic force density as it is for the electric force density.