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.

DC-Field Approximation

The DC electric and magnetic fields from an electric dipole source can be obtained from the full analytic solutions by taking the limit as \(\omega \rightarrow 0\). In this case, the wave-number \(k \rightarrow 0\). For an electric dipole source \(\mathbf{\hat x} I ds\), the DC electric field within a homogenous medium is given by:

(208)\[\lim_{\omega \rightarrow 0} \mathbf{E_e} = \frac{I ds}{4 \pi \sigma r^3} \left[ \left(\frac{3x^2}{r^2} - 1 \right) \mathbf{\hat x} + \frac{3xy}{r^2} \mathbf{\hat y} + \frac{3xz}{r^2} \mathbf{\hat z} \right]\]

According to Eq. (208), the DC electric field depends solely on the observation location and the conductivity of the medium. The source and the electric field are also completely in-phase. Similarly, the corresponding DC magnetic field within the medium is given by:

(209)\[\lim_{\omega \rightarrow 0} \mathbf{H_e} = \frac{I ds}{4 \pi r^2} \left( -\frac{z}{r} \mathbf{\hat y} + \frac{y}{r} \mathbf{\hat z} \right)\]

According to Eq. (209), the DC magnetic field is independent of any physical properties. In addition, the DC electric and magnetic fields are in-phase with one another.

Near-Field Approximation

x For fields which are sufficiently close to the electric dipole source, we may assume that \(| kr | \ll 1\). In this case, the exponential term in \(\mathbf{E}_e\) and \(\mathbf{H}_e\) can be approximated using Taylor expansion:

(210)\[e^{-ikr} \approx 1 - ikr + O \left ( k^2 r^2 \right )\]

The near-field approximation for \(\mathbf{E}_e\) can be obtained by replacing the exponential term in the full analytic solution with the Taylor series approximation from Eq. (210). Thus:

(211)\[\begin{split}\begin{split} \mathbf{E_e} \approx \frac{I ds}{4 \pi (\sigma + i \omega \varepsilon) r^3} \bigg ( 1 - ikr + & O \big ( k^2 r^2 \big ) \bigg ) \Bigg [ \Bigg ( \frac{x^2}{r^2} \mathbf{\hat x} + \frac{xy}{r^2} \mathbf{\hat y} + \frac{xz}{r^2} \mathbf{\hat z} \Bigg ) ... \\ &\big ( -k^2 r^2 + 3ikr +3 \big ) + \big ( k^2 r^2 - ikr -1 \big ) \mathbf{\hat x} \Bigg ] \end{split}\end{split}\]

Eq. (211) can be simplified by neglecting polynomial terms which are \(O(k^2 r^2)\) or higher. Assuming we are in the quasi-static regime (\(| \omega\varepsilon | \ll \sigma\)), the electric field in close proximity to an electric dipole moment \(\mathbf{\hat x} I ds\) is given by:

(212)\[\mathbf{E_e} \approx \frac{I ds}{4 \pi \sigma r^3} \left[ \left(\frac{3x^2}{r^2} - 1 \right) \mathbf{\hat x} + \frac{3xy}{r^2} \mathbf{\hat y} + \frac{3xz}{r^2} \mathbf{\hat z} \right] + O(k^2 r^2 )\]

According to Eq. (212), the near electric field depends only on the observation location and the conductivity of the medium. Additionally, the source and the electric field are completely in-phase.

The near-field approximation for \(\mathbf{H}_e\) can be obtained by replacing the exponential term in the full analytic solution with the Taylor series approximation from Eq. (210). Thus:

(213)\[\mathbf{H_e} \approx \frac{I ds}{4 \pi r^2} \left( ikr + 1 \right ) \bigg ( 1 - ikr + O \big ( k^2 r^2 \big ) \bigg ) \left( -\frac{z}{r} \mathbf{\hat y} + \frac{y}{r} \mathbf{\hat z} \right)\]

Eq. (213) can be further simplified by neglecting polynomial terms which are \(O(k^2 r^2)\) or higher. Therefore, the magnetic field in close proximity to electric dipole moment \(\mathbf{\hat x} I ds\) is approximately equal to:

(214)\[\mathbf{H_e} \approx \frac{I ds}{4 \pi r^2} \left( -\frac{z}{r} \mathbf{\hat y} + \frac{y}{r} \mathbf{\hat z} \right) + O(k^2 r^2 )\]

According to Eq. (214), \(\mathbf{H}_e\) does not depend on the physical properties of the background medium. Furthermore, Eq. (214) indicates that \(\mathbf{E}_e\) and \(\mathbf{H}_e\) are in-phase.

Far-Field Approximation

For fields which are sufficient far away from the electric dipole source, we may assume that \(1 \ll | kr |\). In this case, Taylor expansion may not be used to simplify exponential terms in full analytic solutions for the fields. Expressions may still be simplified, however, by considering the largest order terms in each equation.

Let us first consider the far-field approximation of \(\mathbf{E}_e\) within a uniform medium. For off-axis locations (\(y,z \not \ll x\)), only \(O (k^2r^2)\) terms are needed to accurately approximate the electric field from an electric dipole source. However, in the case where (\(y,z \ll x\)), second order terms in the \(\mathbf{\hat x}\) direction cancel, and both the \(\mathbf{\hat y}\) and \(\mathbf{\hat z}\) are insignificant due to geometry. Assuming we are in the quasi-static regime (\(|\omega\varepsilon | \ll \sigma\)), and given that \(k^2 = - i \omega \mu \sigma\), the far field approximation of \(\mathbf{E}_e\) is represented by the following two cases:

\[\begin{split}\mathbf{E_e} \approx \begin{cases} \dfrac{i\omega \mu I ds}{4 \pi r} e^{-ikr} \Bigg [ \left ( \dfrac{x^2}{r^2} - 1 \right ) \mathbf{\hat x} + \dfrac{xy}{r^2} \, \mathbf{\hat y} + \dfrac{xz}{r^2} \, \mathbf{\hat z} \Bigg ] \; \; &\textrm{for} \; \; y,z \not \ll x \\ \; & \; \\ \dfrac{ik Ids}{2\pi \sigma x^2} e^{-ikx} \mathbf{\hat x} &\textrm{for} \; \; y,z \ll x \end{cases}\end{split}\]

Let us now consider the far-field approximation of \(\mathbf{H}_e\) within a uniform medium. Since \(1 \ll | kr |\), we can simplify the full analytic expression in the same manner and show that:

\[\mathbf{H_e} \approx \frac{ik I ds}{4\pi r} e^{-ikr} \left ( -\frac{z}{r} \mathbf{\hat y} + \frac{y}{r}\mathbf{\hat z} \right )\]