# Derivation of the Excitation Factor

Purpose

In this section, the excitation factor for a conductive and magnetically permeable sphere in free-space is derived according to Wait ([Wai51]).

The geometry of this problem is illustrated in Fig. 126, where $${\bf H_0}(i\omega)$$ represents the inducing field, $${\bf m}(i\omega)$$ is the induced magnetic dipole moment of the sphere, and $${\bf H}(i\omega)$$ is the resulting dipole response. $$\sigma_s$$, $$\mu_s$$ and $$\varepsilon_s$$ represent the conductivity, magnetic permeability and electric permittivity of the sphere, respectively. $$\sigma_b$$, $$\mu_b$$ and $$\varepsilon_b$$ represent the conductivity, magnetic permeability and electric permittivity of the background media. Fig. 126 Dipolar response $${\bf H} (i\omega)$$ (purple) from a conductive and magnetically permeable sphere, under the influence of a spatially uniform, harmonic inducing field $${\bf H_0} (i\omega)$$ (red). The induced dipole moment (blue) is represented by $${\bf m} (i\omega)$$.

Let us begin the derivation by considering Maxwell’s equations in the frequency domain:

(393)\begin{split}\begin{align} \nabla \times {\bf E} &= - i \omega \mu {\bf H}\\ \nabla \times {\bf H} &= \big ( \sigma + i \omega \varepsilon \big ) {\bf E} \end{align}\end{split}

where $$\omega$$ is the angular frequency, $$\sigma$$ is the conductivity, $$\mu$$ is the magnetic permeability, and $$\varepsilon$$ is the electric permittivity. According to Ward and Hohmann (1988), the electric field $${\bf E}$$ and magnetic field intensity $${\bf H}$$ may be written in terms of the following Schelkunoff potential $${\bf F}$$, where:

(394)${\bf E} = - \nabla \times {\bf F}$

and

(395)${\bf H} = - \big (\sigma + i \omega \varepsilon \big ) {\bf F} + \frac{1}{i \omega \mu} \nabla \big ( \nabla \cdot {\bf F} \big )$

By subsituting Eqs. (394) and (395) into Maxwell’s equations, we can obtain a wave equation in terms of the $${\bf F}$$ potential:

(396)$\nabla^2 {\bf F} - \gamma^2 {\bf F} = 0$

The wavenumber $$\gamma$$ depends on the physical properties of the media, and is given by:

(397)$\gamma = \Big [ i \omega \mu \sigma - \omega^2 \mu \varepsilon \Big ]^{1/2}$

For his derivation, Wait ([Wai51]) considered the induced magnetic dipole moment resulting from an incident plane wave. If the wavelength of the incident wave is sufficiently larger than the radius of the sphere (i.e. $$|\gamma_b |/2\pi \ll R$$), then we may assume the magnetic field which excites the sphere is spatially uniform about the sphere. For an inducing field of the form $${\bf H_0} (i\omega) = H_0 e^{i\omega t} {\bf \hat z}$$, symmetry of the problem implies that $${\bf E}$$ only has components in $$\boldsymbol{\hat \phi}$$. Therefore by Eq. (394), it follows that our Schelkunoff potential will only have components in $${\bf \hat z}$$ as well.

The Schulkunoff potential may be obtained by considering seperate solutions inside and outside of the sphere:

(398)$\begin{split}{\bf F} (\omega) = \begin{cases} F_b e^{i \omega t} {\bf \hat z} \; \; \textrm{ at } \; \; r>R \\ \\ F_s e^{i \omega t} {\bf \hat z} \; \; \textrm{ at } \; \; r<R \end{cases}\end{split}$

For our problem, boundary conditions on the sphere require that tanjential components of the magnetic field and normal components of the flux density must be continuous. According to Wait ([Wai51]), these conditions are satisfied by the following expressions:

(399)$\begin{split}\textrm{At }r=R: \; \begin{cases} \dfrac{1}{r} \dfrac{\partial F_b}{\partial r} - \gamma_b^2 F_b = \dfrac{1}{r} \dfrac{\partial F_s}{\partial r} - \gamma_s^2 F_s \\ \\ \mu_b \Bigg ( \dfrac{\partial^2 F_b}{\partial r^2} - \gamma_b^2 F_b \Bigg ) = \mu_s \Bigg ( \dfrac{\partial^2 F_s}{\partial r^2} - \gamma_s^2 F_s \Bigg ) \end{cases}\end{split}$

To solve the boundary value problem, Wait ([Wai51]) expressed the solutions, both inside and outside of the sphere, as a sum of spherical harmonic modes with coefficients $$a_n$$ and $$b_n$$, respectively. For the boundary conditions to be satisfied however, he found that coefficients $$a_n=b_n=0 \; \forall \; n>0$$. As a result, the solution to the Schelkunoff potentials inside and outside the sphere are defined by:

(400)$F_b = - \frac{H_0 }{\sigma_b + i \omega \varepsilon_b} + i \omega \mu_b \frac{e^{-\gamma_b r}}{r}a_0 H_0$

and

(401)$F_s = i \omega \mu_s \frac{sinh \big ( \gamma_s r \big )}{r} b_0 H_0$

To determine the solution outside of the sphere, Eqs. (400) and (401) may be substituted into Eq. (399). Through meticulous algebra, coefficient $$a_0$$ can be expressed as:

(402)$a_0 \! =\! \frac{R^3}{2 e^{-\alpha_b}} \!\Bigg [ \! \frac{2\mu_s \big [ tanh(\alpha_s) - \alpha_s \big ] + \mu_b \big [\alpha_s^2 \, tanh(\alpha_s) - \alpha_s + tanh(\alpha_s) \big ] }{\mu_s \big ( \alpha_b^2 +\alpha_b + 1 \big ) \big [ tanh(\alpha_s) - \alpha_s \big ] - \mu_b \big ( \alpha_b + 1 \big ) \big [ \alpha_s^2 \, tanh(\alpha_s) - \alpha_s + tanh(\alpha_s) \big ] } \! \Bigg ]$

where

(403)$\alpha_b = \gamma_b R = \Big [ i \omega \mu_b \sigma_b - \omega^2 \mu_b \varepsilon_b \Big ]^{1/2} R$

and

(404)$\alpha_s = \gamma_s R = \Big [ i \omega \mu_s \sigma_s - \omega^2 \mu_s \varepsilon_s \Big ]^{1/2} R$

The total magnetic field outside the sphere, in response to an inducing field of the form $${\bf H_0} e^{i\omega t}$$, may be obtained by substituting Eqs. (400) and (402) into Eq. (395). Note that our derivation of $$a_0$$ did not require us to include the frequency-dependent term $$e^{i\omega t}$$ of the primary field. Therefore, we may generalize our solution for any harmonic inducing field of the form $${\bf H_0} (i\omega )$$.

If the sphere lies within a resistive background ($$\sigma_b \ll \sigma_s$$, $$\mu_b = \mu_0$$, and $$\varepsilon_b = \varepsilon_0$$), and if electric displacement within the sphere is neglected ($$\omega \varepsilon_s \ll \sigma_s$$), then Eq. (402) reduces to:

(405)$a_0 \! =\! \frac{R^3}{2} \!\Bigg [ \! \frac{2\mu_s \big [ tanh(\alpha) - \alpha \big ] + \mu_0 \big [\alpha^2 \, tanh(\alpha) - \alpha + tanh(\alpha) \big ] }{\mu_s \big [ tanh(\alpha) - \alpha \big ] - \mu_0 \big [ \alpha^2 \, tanh(\alpha) - \alpha + tanh(\alpha) \big ] } \! \Bigg ]$

where

(406)$\alpha = \Big [ i\omega \mu_s \sigma_s \Big ]^{1/2}R$

Wait ([Wai51]) simplified the solution outside the sphere by considering the dipole field within a vacuum. For a dipole moment $${\bf m} (i\omega)$$, the dipole field $${\bf H} (i\omega)$$ is given by (Griffiths, 1999):

(407)${\bf H} (i \omega) =\frac{1}{4\pi} \Bigg [ \frac{3 {\bf r} \; \big [ {\bf m} (i\omega) \cdot {\bf r} \; \big ]}{r^5} - \frac{{\bf m} (i\omega) }{r^3} \Bigg ]$

where $${\bf r}$$ defines the spatial vector from $$P$$ to $$Q$$. The dipole field was derived by performing a multipole expansion on Eq. (400), and neglecting higher order terms. This lead to an explicit expression for the magnetic dipole moment in terms of coefficient $$a_0$$, where:

(408)${\bf m} (i \omega) = 4 \pi a_0 {\bf H_0} (i \omega) = \frac{4\pi}{3}R^3 \chi (i \omega) {\bf H_0} (i \omega)$

According the Eq. (408), $${\bf m} (i\omega)$$ may also be expressed as the product of the inducing field, the sphere’s volume, and an excitation factor $$\chi (i\omega)$$, where:

(409)$\chi (i \omega) = \frac{3}{2} \Bigg [ \! \frac{2\mu_s \big [ tanh(\alpha) - \alpha \big ] + \mu_0 \big [\alpha^2 \, tanh(\alpha) - \alpha + tanh(\alpha) \big ] }{\mu_s \big [ tanh(\alpha) - \alpha \big ] - \mu_0 [ \alpha^2 \, tanh(\alpha) - \alpha + tanh(\alpha) \big ] } \! \Bigg ]$