$12\mathrm{.}26$ Application: Communication with Trapped Miners. A serious problem with mine accidents is the difficulty of communicating with trapped miners. The main problem is the lossy nature of soils. Properties of soil depend on composition and moisture but may be approximated as ${}_r=3\mathrm{.}2$ and $=0\mathrm{.}02\frac{S}{m}.$ Suppose that one wishes to communicate with miners trapped underground by generating a wave at the surface with an electric field intensity of amplitude $400\frac{V}{m}$ and a frequency of $300$ MHz $.$

$($a$)$ If the minimum electric field intensity required at the receiver is ${10}^{-6}\frac{V}{m},$ what is the maximum depth of communication possible?

$($b$)$ Suppose one tries the same at a frequency $10$ times higher. What is now the maximum depth?

$($c$)$ Suppose one tries the same at a frequency $10$ times lower. What is now the maximum depth?

$($d$)$ What is your conclusion from these calculations regarding the possibility of wireless communication with trapped miners?

Note: There are other effects that limit communication underground, some of which will be discussed in the following chapter including reflections and effects on antennas but these are neglected here.