# Transient Magnetic Dipole

Purpose

Here, we provide a physical description of the time-dependent magetic dipole. This is used to develop a mathematical expression which can be used to replace the magnetic source term in Maxwell’s equations. We then consider a transient magnetic dipole; which represents a more commonly used geophysical source.

General Definition

The time-dependent magnetic dipole can be thought of as an infinitesimally small loop which carries a time-dependent current. The strength of the source is therefore defined by a time-dependent dipole moment $$\mathbf{m}(t)$$. For a time-dependent magnetic dipole defined by surface area vector $$\mathbf{S}$$ and current $$I (t)$$, the dipole moment is given by:

$\mathbf{m}(t) = I(t)\mathbf{S}$

As a result, the source term for the time-depedent magnetic dipole is given by:

$\mathbf{j_m (r)} = - \dfrac{\partial I}{\partial t} \mathbf{S} \, \delta (x) \delta (y) \delta (z)$

where $$\delta (x)$$ is the Dirac delta function. Notice how the source term contains a time-derivative of the loop’s current. By including the source term, Maxwell’s equations in the time domain are given by:

$\begin{split}\begin{split} \nabla \times \mathbf{e_m} + \mu \dfrac{ \mathbf{h_m} }{\partial t} &= - \dfrac{\partial I(t)}{\partial t} \mathbf{S} \, \delta (x) \delta (y) \delta (z) \\ \nabla \times \mathbf{_m} - & \sigma \mathbf{e_m} - \varepsilon \dfrac{\partial \mathbf{e_m} }{\partial t} = 0 \end{split}\end{split}$

where subscripts $$_m$$ remind us that we are considering a magnetic source. The source is responsible for generating a primary magnetic field in the surrounding region (Fig. 74). According to Faraday’s law, time-varying magnetic fields generate rotational electric fields. In matter, this leads to an induced current density which produces secondary magnetic fields according to the Ampere-Maxwell equation.

Transient Electrical Current Dipole

The transient response represents the response of a system to step-off excitation. For a transient magnetic dipole with surface area vector $$\mathbf{S}$$, the electromagnetic response results from a step-off current of the form $$I (t) = I u(-t)$$. Thus the dipole moment is given by:

$\mathbf{m}(t) = I u(-t) \mathbf{S}$

where $$u(t)$$ is the unit step function and $$I$$ is the amplitude of the current at $$t \leq 0$$. The source term for the corresponding magnetic dipole is given by:

$\mathbf{j_m^s} = I \delta(t) \mathbf{S} \, \delta (x) \delta (y) \delta (z)$

where $$\delta (x)$$ is the Dirac delta function. By including the source term, Maxwell’s equations in the time domain are given by:

\begin{split}\begin{align} \nabla \times \mathbf{e_m} + \mu \dfrac{\partial \mathbf{h_m} }{\partial t} &= I \delta(t) \mathbf{S} \, \delta (x) \delta (y) \delta (z) \\ \nabla \times \mathbf{h_m} - &\sigma \mathbf{e_m} - \varepsilon \dfrac{\partial \mathbf{e_m} }{\partial t} = 0 \end{align}\end{split}

It is possible to solve this system to obtain analytic solutions for the transient electric and magnetic fields. However, we will apply a different approach which uses the inverse Laplace transform.

Organization

In the following section, we solve Maxwell’s equations for a transient magnetic dipole source and provide analytic expressions for the electric and magnetic fields within a homogeneous medium. Asymptotic expressions are then provided for several cases. Numerical modeling tools are made available for investigating the dependency of the electric and magnetic fields on various parameters.