## 2 Diffusion Characteristics and Proportional Coefficients of Space States
### Proportional Coefficients of Different Space States
According to the investigation of the transient electromagnetic field [18; 19; 20; 21], the analytical solution of the half space magnetic field under the condition of the uniform medium can be obtained, namely:  
\[H_{z}=\frac{M}{4\pi R^{3}}\Bigg{[}\bigg{(}1-\frac{9}{u^{2}}\bigg{)}\varphi(u)+ \sqrt{\frac{2}{\pi}}\mathrm{e}^{-(u^{2}/2)}\bigg{(}\frac{9}{u}+2u\bigg{)} \Bigg{]} \tag{10}\]  
According to Formulae (9) and (10), the proportional coefficient of full space to half space can be obtained, and the same coefficient term can be removed, namely:  
\[k_{1}=2\frac{\bigg{[}\varphi(u)-\sqrt{\frac{2}{\pi}}u\mathrm{e}^{-(u^{2}/2)} \bigg{]}}{\bigg{[}\bigg{(}1-\frac{9}{u^{2}}\bigg{)}\varphi(u)+\sqrt{\frac{2}{ \pi}}\mathrm{e  
substituted into Formula (11) to obtain the proportional coefficient of full space to half space in the early stage, as shown in Formula (12).  
\[k_{1}=2\frac{\left[\varphi(u)-\sqrt{\frac{2}{\pi}}u\mathrm{e}^{-(u^{2}/2)}\right] }{\left[\left(1-\frac{9}{u^{2}}\right)\varphi(u)+\frac{9}{u}\sqrt{\frac{2}{\pi}} \mathrm{e}^{-(u^{2}/2)}+2u\sqrt{\frac{2}{\pi}}\mathrm{e}^{-(u^{2}/2)}\right]}=2 \frac{(1-0)}{(1+0+0)}=2 \tag{12}\]  
In the late stage of transient electromagnetic field, \(u\) is a small parameter (this is equivalent to the late stage, the distance between the observation point and the field source is short or the conductivity of the medium is low), the probability integral \(\varphi(u)\) and the exponential function \(\mathrm{e}^{-(u^{2}/2)}\) can be expanded into a series of \(u\), as shown in Formula (13). According to the series theory, the first three terms are taken as [PERSON]'s expansions of probability integral \(\varphi(u)\) and exponential function \(\mathrm{e}^{-(u^{2}/2)}\), as shown in Formula (14). \[\mathrm{e}^{-(u^{2}/2)}=1-\frac{u^{2}}{2}+\frac{u^{4}}{8}-R_{n+1}(u) \tag{13}\]  
\[\varphi(u)=\sqrt{\frac{2}{\pi}}\int_{0}^{u}e^{-(t^{2}/2)}\mathrm{d}t=\sqrt{ \frac{2}{\pi}}\bigg{(}u-\frac{u^{3}}{6}+\frac{u^{5}}{40}-P_{n+1}(u)\bigg{)} \tag{14}\]  
where \(R_{n+1}(u)\) and \(P_{n+1}(u)\) are the high-order remainder of the exponential function \(\mathrm{e}^{-(u^{2}/2)}\) and the probability integral \(\varphi(u)\) respectively, and \(n\) is 2. By abandoning the higher-order error term and substituting Formulas (13) and (14) into Formula (11), the proportional coefficient of full space to half space in the late period can be obtained, as shown in Formula (15):  
\[ \begin{array}{c}k_{1}=2\frac{\sqrt{\frac{2}{\pi}}\left(\left(u-\frac{u^{3}}{ 6}+\frac{u^{5}}{6}\right)-\sqrt{\frac{2}{\pi}}\left(u-\frac{u^{3}}{6}+\frac{u ^{5}}{6}\right)\right.}{\left[\left(1-\frac{9}{u^{2}}\right)\sqrt{\frac{2}{ \pi}}\left(u-\frac{u^{3}}{6}+\frac{u^{5}}{6}\right)+\sqrt{\frac{2}{\pi}}\left( 1-\frac{u^{2}}{2}+\frac{u^{4}}{8}\right)\left(\frac{2}{\pi}+2u\right)\right]} }\\ =2\frac{\left(\frac{u^{3}}{6}-\frac{u^{5}}{10}\right)}{\left(\frac{u^{3}}{15}- \frac{115}{40}\right)}\approx 2\frac{\frac{u^{3}}{45}}{\frac{u^{3}}{45}}=2.5\end{array}  \tag{15}\]  
Formula (12) shows that the induced current under the uniform medium has not been diffused when the current is just disconnected. The proportional coefficient of full space to half space is 2.0, theoretically. It can be considered that the electromagnetic field in the full space state is the superposition of the electromagnetic fields in the upper and lower half spaces. According to Formula (15), the proportional coefficient of full space to half space is 2.5, theoretically, during the induced current diffusion. The electromagnetic field in the full space state is not only the superposition of the electromagnetic fields in the upper and lower half spaces, but also includes the additional electromagnetic fields caused by the upper and lower half spaces (called the'space effect'). In the numerical simulation, the induced currents in the full space state and the half space state gradually spread outward. The proportional coefficient of full space to half space maintains a slight variation with an average value of 2.5 in the whole process.