Asymptotic Approximations

Purpose

Here, simplified expressions for the electric and magnetic fields are presented for several cases. By examining simplified expressions, we can more easily see how the fields depend on certain parameters. As the full analytic solution for the vector potential is rather simple, asymptotic approximations are not presented here.

Near-Field/Late-Times

For fields which are very close to the electric dipole source, or at sufficiently late times:

(237)\[\theta r = \Bigg ( \frac{\mu \sigma}{4t} \Bigg )^{1/2} r \ll 1\]

In this case, the exponential and error functions can be approximated using Taylor expansion. Thus:

(238)\[e^{-\theta^2 r^2} = 1 - \theta^2 r^2 + \frac{1}{2}\theta^4 r^4 + \; ...\]

and

(239)\[\textrm{erf}(\theta r) = \frac{2}{\sqrt{\pi}} \theta r - \frac{2}{3 \sqrt{\pi}}\theta^3 r^3 + \frac{1}{5\sqrt{\pi}} \theta^5 r^5 + \;...\]

By substituting the above Taylor expansions into the full analytic solutions for \({\bf e_e}\) and \({\bf h_e}\), we can obtain near-field/late-time approximations. In the case of the electric field:

(240)\[{\bf e_e}(t) \approx \frac{ Ids}{15 \pi^{3/2} \sigma r^3} \Bigg [ 6 \,\theta^5 r^5 \Bigg ( \frac{x^2}{r^2}\mathbf{\hat x} + \frac{xy}{r^2}\mathbf{\hat y} + \frac{xz}{r^2}\mathbf{\hat z} \Bigg ) + \Big ( 10 \,\theta^3 r^3 + 3 \,\theta^5 r^5 \Big ) \mathbf{\hat x} \Bigg ]\]

According to Eq. (240), \(\mathbf{\hat y}\) and \(\mathbf{\hat z}\) components of the near-field/late-time electric field decay proportional to \(t^{-5/2}\). However, \(\theta^3 r^3\) terms for the \(\mathbf{\hat x}\) component do not cancel. Therefore, the \(\mathbf{\hat x}\) component of the electric field decays proportional to \(t^{-3/2}\) at sufficiently late times. For the magnetic field, the near-field/late-time approximation is given by:

(241)\[{\bf h_e}(t) \approx \frac{\theta^3 Ids}{3\pi^{3/2}} \big (-z \, \mathbf{\hat y} + y \, \mathbf{\hat z} \big )\]

According to Eq. (241), the near-field/late-time electric field decays proportional to \(t^{-3/2}\). Taking the derivative of Eq. (241), near-field/late-time approximation for the time-derivative of the magnetic field is given by:

(242)\[\frac{\partial {\bf h_e}}{\partial t} \approx \frac{2 \theta^5 Ids}{\mu \sigma \pi^{3/2}} \big ( z \, \mathbf{\hat y} - y \, \hat z \big )\]

According to Eq. (242), the time-derivative of the magnetic field decays proportional to \(t^{-5/2}\).

Far-Field

For fields which are far from the electrical current dipole source, or at sufficiently early times:

(243)\[\theta r = \Bigg ( \frac{\mu \sigma}{4t} \Bigg )^{1/2} r \gg 1\]

In this case, the exponential and complimentary error function can be approximated as follows:

(244)\[e^{-\theta^2 r^2} \approx 0\]

and

(245)\[\textrm{erfc}(\theta r) \approx 0\]

As a result, there are no interesting asymptotic approximations for the far-field.