Answer to Question #173753 in Electricity and Magnetism for chinaza

Question

derive the expression for penetrating depth of magnetotelluric signals

Answer 1

Electric-magnetic phenomena can be represented by the Maxwell's equations. 

$J(t)+\frac{\partial \:}{\partial \:t}\left(D(t\right))$

Solution to Maxwell's equations in simple medium

$E_x = A e^{−kz}+B e^{+kz}$

$H_y = \frac{k}{iω \mu_0}(A e^{−kz}+B e^{+kz})$

The homogenous in 1 D

$E_x = A e^{−kz}$

$k =\sqrt{ iω \mu_0/\rho}$

$H_y = \frac{k}{iω \mu_0}(A e^{−kz})$

At $E_x (z=0)= A \implies E_x(z=\delta) = A/ e$

Therefore, penetrating depth of magnetotelluric signals, $\delta =\sqrt{ \frac{2 \rho}{iω \mu_0}}$

Where $ω= 2 \pi f$