{"action":"created","comment":{"url":"https://api.github.com/repos/ubcgif/em/pulls/comments/53575318","id":53575318,"diff_hunk":"@@ -0,0 +1,262 @@\n+.. _sphere_freespace:\n+\n+Response from a Conducting and Permeable Sphere in Free-Space\n+-------------------------------------------------------------\n+\n+Introduction\n+============\n+\n+Frequency-domain responses from ore bodies and unexploded ordnance items are of great practical importance to geophysicists.\n+On a fundamental level, these responses can be understood by considering the excitation of a conductive and magnetically permeable sphere, under the influence of a harmonic inducing field.\n+Here, expressions derived by Wait (1951) are used to characterize the excitation of the sphere in terms of an induced dipole moment.\n+We examine how a target's dipole moment depends on its size and physical properties.\n+A simple model for predicing the sphere's dipole response in free-space is then presented.\n+For a sufficiently compact target, the free-space response represents a low-order approximation to the response of a conductive and permeable object within a resistive medium.\n+\n+\n+Plane Wave Excitation of a Conducting and Magnetically Permeable Sphere\n+=======================================================================\n+\n+In this section, equations derived by Wait (1951) are used to describe the induced magnetic dipole moment of a conductive and permeable sphere, resulting from excitation by a uniform magnetic field (Figure ??).\n+According to Ward and Hohmann (1989), the frequency-domain wave equation within a homogeneous media can be expressed in terms of the magnetic Schelkunoff potential :math:`\\vec F`:\n+\n+.. figure:: ./images/figGeometrySphere.png\n+\t:align: right\n+        :scale: 45%\n+        :name: GeometrySphere\n+\n+.. math::\n+\t\\nabla^2 \\vec F - \\gamma^2 \\vec F = 0\n+\t:label: Schelkunoff_Wave\n+\n+For a propagation media with conductivity :math:`\\sigma`, magnetic permeability :math:`\\mu` and electric permittivity :math:`\\varepsilon`, the wavenumber :math:`\\gamma` can be expressed as follows:\n+\n+.. math::\n+\t\\gamma = \\Big [ i \\omega \\mu \\sigma - \\omega^2 \\mu \\varepsilon \\Big ]^{1/2}\n+\t:label: Wave_Number\n+\n+By definition of the Schelkunoff potential, the electric field :math:`\\vec E` may be obtained by substituting :math:`\\vec F` into the following expression:\n+\n+.. math::\n+\t\\vec E = - \\nabla \\times \\vec F\n+\t:label: EcurlF\n+\n+And for the magnetic field :math:`\\vec H`:\n+\n+.. math::\n+\t\\vec H = - \\big (\\sigma + i \\omega \\varepsilon \\big ) \\vec F + \\frac{1}{i \\omega \\mu} \\nabla \\big ( \\nabla  \\cdot \\vec F \\big )\n+\t:label: SchelkunoffH\n+\n+For his derivation, Wait (1951) began by considering the induced magnetic dipole moment resulting from an incident plane wave.\n+If the wavelength of the incident wave is sufficiently larger than the radius of the sphere (i.e. :math:`|\\gamma_b |/2\\pi \\ll R`), then we may assume the magnetic field which induces the dipole moment is approximately uniform about the sphere.\n+For an inducing field of the form :math:`\\vec H_0 (i\\omega) = H_0 e^{i\\omega t} \\hat z`, symmetry of the problem implies that :math:`\\vec E` only has components in :math:`\\hat \\phi`.\n+Therefore by Eq. :eq:`EcurlF`, it follows that our Schelkunoff potential will only have components in :math:`\\hat z`.\n+The Schulkunoff potential may be obtained by considering seperate solutions inside and outside of the sphere:\n+\n+.. math::\n+\t\\vec F (\\omega) = \\begin{cases}\n+\tF_b e^{i \\omega t} \\hat z \\; \\; \\textrm{  at  } \\; \\; r>R \\\\\n+\t\\\\\n+\tF_s e^{i \\omega t} \\hat z \\; \\; \\textrm{  at  } \\; \\; r<R \n+\t\\end{cases}\n+\t:label: SolnsInsideOutside\n+\n+\n+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.\n+According to Wait (1951), these conditions are satisfied by the following:\n+\n+.. math::\n+\t\\textrm{At }r=R: \\; \\begin{cases}\n+\t\\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 \\\\\n+\t  \\\\\n+\t\\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 )\n+\t\\end{cases}\n+\t:label: BoundaryConditions\n+\n+To solve the boundary value problem, Wait (1951) expressed the Schelkunoff potentials, both inside and outside of the sphere, as a sum of spherical harmonic modes with coefficients :math:`a_n` and :math:`b_n`, respectively.\n+For the boundary conditions to be satisfied however, he found that coefficients :math:`a_n=b_n=0 \\; \\forall \\; n>0`.\n+As a result, the solution to the Schelkunoff potentials inside and outside the sphere are defined by:\n+\n+.. math::\n+\tF_b = - \\frac{H_0 }{\\sigma_b + i \\omega \\varepsilon_b} + i \\omega \\mu_b  \\frac{e^{-\\gamma_b r}}{r}a_0 H_0\n+\t:label: Foutside\n+\n+and\n+\n+.. math::\n+\tF_s = i \\omega \\mu_s \\frac{sinh \\big ( \\gamma_s r \\big )}{r} b_0 H_0\n+\t:label: Finside\n+\n+To determine the solution outside of the sphere, Eqs. :eq:`Foutside` and :eq:`Finside` may be substituted into Eq. :eq:`BoundaryConditions`.\n+Through meticulous algebra, coefficient :math:`a_0` can be expressed as:\n+\n+.. math::\n+\ta_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 ]\n+\t:label: a0\n+\t\n+where\n+\n+.. math::\n+\t\\alpha_b = \\gamma_b R = \\Big [ i \\omega \\mu_b \\sigma_b - \\omega^2 \\mu_b \\varepsilon_b \\Big ]^{1/2} R\n+\t:label: alpha_b\n+\t\n+and\n+\n+.. math::\n+\t\\alpha_s = \\gamma_s R = \\Big [ i \\omega \\mu_s \\sigma_s - \\omega^2 \\mu_s \\varepsilon_s \\Big ]^{1/2} R\n+\t:label: alpha_s\n+\n+The total magnetic field outside the sphere, in response to an inducing field of the form :math:`\\vec H_0 e^{i\\omega t}`, may be obtained by substituting Eqs. :eq:`Foutside` and :eq:`a0` into Eq. :eq:`SchelkunoffH`.\n+Note that our derivation of :math:`a_0` did not require us to include the frequency-dependent term :math:`e^{i\\omega t}` of the primary field.\n+Therefore, we may generalize our solution for any inducing field of the form :math:`\\vec H_0 (i\\omega )`.\n+For practical purposes, it is common to examine the dipole response of the sphere.\n+In this case, the dipole response :math:`\\vec B (\\omega)` at location :math:`Q` is:\n+\n+.. math::\n+\t\\vec B (\\omega) =\\frac{\\mu_0}{4\\pi} \\Bigg [ \\frac{3\\vec r \\; \\big [ \\vec m(\\omega) \\cdot \\vec r \\; \\big ]}{r^5} - \\frac{\\vec m (\\omega) }{r^3} \\Bigg ] \n+\t:label: DipoleField\n+\n+where :math:`\\mu_0` is the permeability of free-space, :math:`\\vec r` defines the spatial vector from :math:`P` to :math:`Q`, and :math:`\\vec m (\\omega)` is the frequency-dependent dipole moment induced by the primary field.\n+The dipole moment can be expressed as the product of the sphere's volume, the inducing field, and a magnetization factor :math:`\\chi (\\omega)`:\n+\n+.. math::\n+\t\\vec m (\\omega) = 4 \\pi a_0 \\vec H_0 (i \\omega) = \\frac{4\\pi}{3}R^3 \\chi (\\omega) \\vec H_0 (i \\omega)\n+\t:label: DipoleMoment\n+\n+where\n+\n+.. math::\n+\t\\chi (\\omega) \\! =\\! \\frac{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 ]\n+\t:label: ChiFull\n+\n+If the sphere is located within a resistive medium, then :math:`\\alpha_b \\ll \\alpha_s`, :math:`\\mu_b = \\mu_0`, and Eq. :eq:`ChiFull` will reduce to:\n+\n+.. math::\n+\t\\chi (\\omega) = \\frac{3}{2} \\Bigg [ \\! \\frac{2\\mu_s \\big [ tanh(\\alpha_s) - \\alpha_s  \\big ] + \\mu_0 \\big [\\alpha_s^2 \\, tanh(\\alpha_s) - \\alpha_s + tanh(\\alpha_s) \\big ] }{\\mu_s  \\big [ tanh(\\alpha_s) - \\alpha_s \\big ] - \\mu_0 [ \\alpha_s^2 \\, tanh(\\alpha_s) - \\alpha_s + tanh(\\alpha_s) \\big ] } \\! \\Bigg ]\n+\t:label: ChiApprox\n+\n+\n+.. figure::\n+\t./images/figChiOmega.png\n+\t:figwidth: 40%\n+\t:align: right\n+\n+\tMagnetization factor for a sphere in free-space with :math:`\\sigma_s` = 10 S/m, :math:`\\mu_s` = 1.1 :math:`\\mu_0` , :math:`\\varepsilon_s` = :math:`\\varepsilon_0`, and :math:`R` = 25 m.\n+\t\t\n+.. figure::\n+\t./images/figChiKappa.png\n+\t:figwidth: 40%\n+\t:align: right\n+\t\n+\tMagnetization facter at :math:`\\omega` = 0 for increasing magnetic susceptibilities (red), compared to a linear trend with respect to :math:`\\kappa` (black).\n+\n+The magnetization factor for a sphere in free space, with :math:`\\sigma_s` = 10 S/m, :math:`\\mu_s` = 1.1 :math:`\\mu_0` , :math:`\\varepsilon_s` = :math:`\\varepsilon_0` and :math:`R` = 25 m, can be seen in Figure (reference).\n+Near the resistive limit, :math:`\\chi (\\omega)` is positive, implying that excitation of the sphere is parallel to the inducing field.\n+Because the EM induction is negligible at sufficiently low frequencies, this case represents a purely magnetic response by the sphere.\n+Near the inductive limit, :math:`\\chi(\\omega)` is negative.\n+Therefore, inductive excitation of the sphere will oppose the inducing field.\n+For a conductive and permeable sphere in free-space, Eq. :eq:`ChiApprox` can be used to show that:\n+\n+.. math::\n+\t\\chi (\\omega) = \\begin{cases}\n+\t\\dfrac{3 \\big ( \\mu_s - \\mu_0 \\big )}{ \\big ( \\mu_s + 2 \\mu_0 \\big ) } = \\dfrac{3 \\kappa }{3 + \\kappa} & \\textrm{ as } \\omega \\rightarrow 0 \\\\\n+\t\\\\\n+\t\\dfrac{3}{2}  & \\textrm{ as } \\omega \\rightarrow \\infty\n+\t\\end{cases}\n+\t:label: ChiLimits\n+\t\n+where :math:`\\kappa` is the magnetic susceptibility of the sphere, and :math:`\\mu_s =\\mu_0 \\big [ 1 + \\kappa \\big ]`.\n+According to Eq. :eq:`ChiLimits`, the inductive limit of :math:`\\chi (\\omega)` is constant.\n+As a result, the dipole moment which characterizes the sphere in this case is proportional only to :math:`R^3`, and the strength of the inducing field.\n+For purely magnetic responses however, the magnetization factor ultimately depends on the magnetic susceptibility of the sphere.\n+For small magnetic susceptibilities (:math:`\\kappa < 0.1`), the relationship between :math:`\\kappa` and the resulting dipole moment is approximately linear.\n+For large values however, the effects of self-demagnetization within the sphere will result in a weaker magnetic dipole moment for the sphere.\n+As :math:`\\kappa \\rightarrow \\infty`, Eq. :eq:`ChiLimits` can be used to show that :math:`\\chi \\rightarrow 3`.\n+The magnetization factor for :math:`\\omega = 0`, denoted here as :math:`\\chi (\\kappa)`, is plotted in Figure (reference). \n+\n+For a purely conductive object (i.e. :math:`\\mu_s = \\mu_0`), Eq. :eq:`ChiApprox` can be further simplified.\n+By neglecting electric displacement (i.e. :math:`\\omega \\varepsilon_s \\ll \\sigma_s`):\n+\n+.. math::\n+\t\\chi (\\omega) = - \\; \\frac{3}{2} \\Bigg [ 1 + \\frac{3}{\\alpha^2} - \\frac{3 \\, \\textrm{coth}(\\alpha)}{\\alpha} \\Bigg ]\n+\t:label: ChiConductive \n+\n+where\n+\n+.. math::\n+\t\\alpha = \\Big [ i \\omega \\mu_0 \\sigma_s \\Big ]^{1/2} R\n+\t:label: alpha_c\n+\t\n+Eq. :eq:`ChiConductive` is commonly used to understand the excitation of a compact conductive object in a resistive medium.\n+\n+\n+\n+Dipole Response in Free-Space\n+=============================\n+\n+Here, we present a model for predicting the anomalous frequency-domain response in free-space, from a conductive and permeable sphere, due to an inductive loop source.\n+This scenario represents a low-order approximation for a conductive and magnetically permeable object within a resistive medium.\n+The geometry of this problem is shown in Figure 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