# 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 and Spies ([WS69]). These expressions can be used to explore the nature of time-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 time-domain response can be understood by considering the problem geometry illustrated in Fig. 139; where the target has conductivity \(\sigma\) and magnetic permeability \(\mu\), and the host medium has conductivity \(\sigma_b\) and magnetic permeability \(\mu_b = \mu_0\). In this case, the transmitter (\(Tx\)) generates a time-dependent primary field \(\mathbf{h_p} (t)\), which induces an excitation within the target. The excitation induced within the target produces a secondary field \(\mathbf{h_s} (t)\), which is then measured by a receiver coil (\(Rx\)).

On a fundamental level, the geophysical problem illustrated in Fig. 139 may be understood by considering a conductive and magnetically permeable sphere in free-space (Fig. 140). This is accomplished by assuming the 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 \(\mathbf{h_0} (t)\) may be calculated using the Biot-Savart law. The free-space problem will act as the basis for our analysis of time-domain responses.

## Outline¶

The analysis presented in this section follows expressions derived by Wait and Spies ([WS69]). Wait and Spies considered the free-space dipole response from a conductive and magnetically permeable sphere, under the influence of a spatially uniform field. Our analysis of the sphere’s time-domain response is presented in four parts:

1. General Formulation: Here, a general formulation for the sphere’s time-domain response is presented according to Wait and Spies ([WS69]). These expressions characterize the sphere’s dipole response under the influence of a spatially uniform time-dependent field.

2. Transient Response: In many cases, we are interested in the transient or “step-off” response from a target body. This is a fundamental response of great importance in geophysics. Here, we examine the sphere’s transient response and how it depends on the sphere’s physical properties.

3. Impulse Response: The sphere’s time-dependent response to an arbitrary primary field is characterized by its impulse response. Here, the impulse response for a permeable and a non-permeable sphere are discussed.

4. Analytic Derivation: Here, abbreviated derivations for the sphere’s step-response and impulse response are presented for a conductive and magnetically permeable sphere. These derivations can be found in Spies ([Wai51]) and Wait and Spies ([WS69]).

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