# Sphere¶

Purpose

Here, we consider the dipole response from a conductive and magnetically permeable sphere in free-space. Analytic expressions presented here were derived by Wait ([Wai51]). These expressions can be used to explore the nature of frequency-domain responses generated by compact ore-bodies and unexploded ordnance items within resistive media. Thus by understanding the response from a sphere, we can gain insight regarding practical geophysical scenarios.

## Introduction¶

For a compact target within a resistive medium, the frequency-domain response can be understood by considering the problem geometry illustrated in Fig. 121; where the target has conductivity \(\sigma\) and magnetic permeability \(\mu\), and the host medium has conductivity \(\sigma_b \ll \sigma\) and magnetic permeability \(\mu_b=\mu_0\). In this case, the transmitter generates a harmonic primary field \({\bf H_p} (i\omega)\), which induces an excitation within the target. The excitation induced within the target produces a secondary field \({\bf H_s} (i\omega)\), which is then measured by a receiver coil (Rx).

On a fundamental level, the geophysical problem illustrated in Fig. 121 may be understood by considering a conductive and magnetically permeable sphere in free-space (Fig. 122). This is accomplished by assuming the frequency-dependent attenuation (link) of EM signals, and inductive responses from the host media, are negligible. For the free-space problem, background physical properties are now given by \(\sigma_b=0\) and \(\mu_b=\mu_0\), and the primary field \({\bf H_0} (i\omega)\) may be calculated using the Biot-Savart law. The free-space problem will act as the basis for our analysis of frequency-domain responses.

## Outline¶

The analysis presented in this section follows expressions derived by Wait ([Wai51]). Wait considered the free-space dipole response from a conductive and magnetically permeable sphere, under the influence of a spatially uniform, harmonic field. Analytic expression, and the conditions in which they are valid, are presented here. The characteristic response of a conductive and magnetically permeable sphere may be defined in terms of an excitation factor. The excitation factors for several special cases are presented here . These cases include: a conductive and magnetically permeable sphere, a purely conductive sphere, and the zero-frequency excitation of a highly permeable sphere. A supporting derivation for the excitation factor can be found here.

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