Calculation of true thickness of the bed

GATE 2018

76) A plane electromagnetic (EM) wave traveling vertically downwards with a frequency of 1000Hz in a homogeneous medium has a skin depth of 100m. The ratio of the amplitude of the EM wave at a depth of 75m with respect to the amplitude at the Earth’s surface is ______________

(Thanks to Neeraja, AU)

Solution:

$A(Z) = A_o e^{-\beta Z} -----(1)$

A(Z) – amplitude of EM at a depth Z

$A_o – amplitude of EM wave at Earth’s surface(Z=0)$ 

$\beta – decay constant$

Z – depth

$The relation between the skin depth(\delta) and decay constant (\beta) is$
 

$\delta = \frac{1}{\beta}-----(2)$

$Skin depth, \delta = 503.8\sqrt{\frac{\rho}{f}}-----(3)$

$\beta = \frac{1}{\delta}=\frac{1}{100}=0.01$

Z=75m

Substitute in eq(1)


$\frac{A_{(75)}}{A_o}= e^{-\beta Z}= e^{-0.01 \times 75} = e^{-0.75}$ 

$\frac{A_{(75)}}{A_o}=0.472$

$\therefore the ratio of the amplitudes= 0.472$