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 magnetic dipole source, or at sufficiently late times:

(300)$$\theta r=(\frac{\mu \sigma }{4t}{)}^{1/2}r\ll 1$$

As a result, the exponential and error functions can be approximated using Taylor expansion, where:

(301)$$e^{-{\theta }^2{r}^2}\approx 1-{\theta }^2{r}^2+\frac{1}{2}{\theta }^4{r}^4+...$$

and

(302)$$\text{erf}(\theta r)=\frac{2}{\sqrt{\pi }}\theta r-\frac{2}{3\sqrt{\pi }}{\theta }^3{r}^3+\frac{1}{5\sqrt{\pi }}{\theta }^5{r}^5+...$$

By substituting the above Taylor series into the analytic expressions for ${\mathbf{e}}_{\mathbf{m}}$ and ${\mathbf{h}}_{\mathbf{m}}$, we can obtain near-field/late-time approximations. In the case of the electric field, the near-field/late-time approximation is given by:

(303)$${\mathbf{e}}_{\mathbf{m}}(t)\approx \frac{2m{\theta }^5}{{\pi }^{3/2}\sigma }(-z\hat{\mathbf{y}}+y\hat{\mathbf{z}})$$

According to Eq. (240), the near-field/late-time electric field decays proportional to $t^{-5/2}$. For the magnetic field, the near-field/late-time approximation is given by:

(304)$${\mathbf{h}}_{\mathbf{m}}(t)\approx \frac{2m}{15{\pi }^{3/2}{r}^3}[3{\theta }^5{r}^5(\frac{x^2}{r^2}\hat{\mathbf{x}}+\frac{xy}{r^2}\hat{\mathbf{y}}+\frac{xz}{r^2}\hat{\mathbf{z}})+(5{\theta }^3{r}^3-6{\theta }^5{r}^5)\hat{\mathbf{x}}]$$

According to Eq. (241), the $\hat{\mathbf{y}}$ and $\hat{\mathbf{z}}$ components of the magnetic field decay proportional to $t^{-5/2}$. For the $\hat{\mathbf{x}}$ component however, ${\theta }^3{r}^3$ terms remain. As a result, the $\hat{\mathbf{x}}$ component of the field decays proportional to $t^{-3/2}$ after sufficient time. The near-field/late-time approximation for the time-derivative of the magnetic field is given by:

(305)$$\frac{\partial {\mathbf{h}}_{\mathbf{m}}}{\partial t}\approx -\frac{4m{\theta }^5}{{\pi }^{3/2}\mu \sigma }[{\theta }^2{r}^2(\frac{x^2}{r^2}\hat{\mathbf{x}}+\frac{xy}{r^2}\hat{\mathbf{y}}+\frac{xz}{r^2}\hat{\mathbf{z}})+(1-2{\theta }^2{r}^2)\hat{\mathbf{x}}]$$

According to Eq. (242), $\hat{\mathbf{y}}$ and $\hat{\mathbf{z}}$ components of the field decay proportional to $t^{-7/2}$. In the $\hat{\mathbf{x}}$ however, ${\theta }^5{r}^5$ terms remain. As a result, the $\hat{\mathbf{x}}$ component of the field decays proportional to $t^{-5/2}$ after sufficient time.

Far-Field

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

(306)$$\theta r=(\frac{\mu \sigma }{4t}{)}^{1/2}r\gg 1$$

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

(307)$$e^{-{\theta }^2{r}^2}\approx 0$$

and

(308)$$\text{erfc}(\theta r)\approx 0$$

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