Subsurface Detection 2 – System Geometry

Let’s assume an ideal and homogeneous half-space. The electromagnetic fields created from a magnetic dipole have cylindrical symmetry about the dipole axis within the media. The dipole is placed at the origin and directed along the z-axis as shown in Figure 1 with the resulting field vectors as shown. Vector $B_\phi(z)$ is measured; it is the distortion of this vector which contains the desired information. The magnetic dipole is formed with a current loop. The geometry of a possible anomaly is illustrated here, but the discussion of secondary fields is deferred for now.

The x-y plane elements are distorted in the illustration for clarity. The y-axis is normal to the illustration.

The far-field free-space magnetic dipole are valid for $\rho(x,y) \; = \; \left(x^2 \, + \, y^2\right)^{1/2} \; >> \, a$ where “ $a$ ” is the transmit coil radius.

$\begin{displaymath} \begin{align} B_\rho \; &= \; \mu_o \, \dfrac{M}{\, 2 \, \pi \, \rho^3 \,} \, cos \, \theta \; \Rightarrow \; \left. 0 \right|_{\theta=\pi/2} \\ \\ B_\theta \; &= \; \mu_o \, \dfrac{M}{\, 4 \, \pi \, \rho^3 \,} \, sin \, \theta \; \Rightarrow \; \left. \mu_0 \, \dfrac{M}{\, 4 \, \pi \, \rho^3 \,} \right|_{\theta=\pi/2} \end{align} \end{displaymath}$

From $B_z \; = \; B_\rho \, cos \. \theta \; - \; B_theta \, sin \theta$ , the vertical field component is expressed as:

$\begin{displaymath} B_z^o \; = \; \mu_o \, \dfrac{M}{\, 4 \, \pi \, \rho^3 \,} \, cos \left(\, 3 \, cos^2 \theta \, - \, 1 \,\right) \; \Rightarrow \; \left. -\mu_0 \, \dfrac{M}{\, 4 \, \pi \, \rho^3 \,} \right|_{\theta=\pi/2} \end{displaymath}$

where $M$ is the transmitted source field represented as a magnetic dipole.

Defining $M$ in the time-domain as $M(t) \, = \, M_o \, sin \, \omega \, t$ , the following expressions define the primary magnetic field for a homogeneous media in which $\sigma \, >> \, \varepsilon \, \omega^*$

$\begin{displaymath} \begin{align} B_R^o(t) \; &= \; \mu_o \, \dfrac{M}{\, 2 \, \pi \, \rho^3 \,} \, cos \, \theta \; sin \, \omega \, t \\ \\ B_\theta^o(t) \; &= \; \mu_o \, \dfrac{M}{\, 4 \, \pi \, \rho^3 \,} \, sin \, \theta \; sin \, \omega \, t \end{align} \end{displaymath}$

A vortex electric field rotates about the dipole axis as:

$\begin{displaymath} E_\phi^o(t) \; = \; \dfrac{\mu_o}{\, 4 \, \pi \, \rho^2 \,} \dfrac{\,\partial \, M_o(t) \,}{\partial \, t} \; \Rightarrow \; \left. - \, M_o \,\dfrac{\, \mu_o \, \omega \, cos \, \omega \, t \,}{\, 4 \, \pi \, \rho^3 \,} \right|_{\theta=\pi/2} \end{displaymath}$

where $M_o(t) \; = \; \pi \, a^2 \, n \, I_o(t)$ which are the source dipole parameters.

The $sin(\omega \, t)$ and $cos(\omega \, t)$ terms are separated for clarity, but arise from $I_o(t) \; = \; I_o \, sin(\omega \, t)$ . Equipotential force contours of the magnetic field component $B_z$ and the electric field component $E_\phi$ are shown for a single quadrant cross-section of free-space in Figure 2.

These field lines represent equipotential contours within a homogeneous half-space created by a surface magnetic dipole M.

Part 1
Part 3

That’s good for now.

$^*$ This assumption implies that conductivity $\sigma$ is more significant than the relative dielectric constant $\varepsilon$ ; i.e., the conduction currents are much greater than displacement currents.

This assumption implies that conductivity is more significant than the relative dielectric constant ; i.e., the conduction currents are much greater than displacement currents.

$^*$ This assumption implies that conductivity $\sigma$ is more significant than the relative dielectric constant $\varepsilon$ ; i.e., the conduction currents are much greater than displacement currents.