Question

If the depth of penetration of an inducing electromagnetic wave of 1 kHz frequency is 400 m, then the resistivity of the subsurface is __________ \(\Omega\)m (answer in integer).

Hint:

Use the EM skin-depth (depth of penetration) relation, \(\delta = 503\sqrt{\rho/f}\), and solve for \(\rho\).

Answer 1

Correct Answer: 630

The depth of penetration of a plane EM wave into a conducting Earth is its skin depth, \(\delta\), given by the standard geophysical rule-of-thumb formula \(\delta = 503\sqrt{\rho/f}\), with \(\delta\) in metres, \(\rho\) in \(\Omega\)m and \(f\) in Hz.

Step 1: Write down the knowns.
\(\delta = 400\) m, \(f = 1000\) Hz.

Step 2: Solve for \(\rho\).
\(400 = 503\sqrt{\rho/1000}\)
\(\sqrt{\rho/1000} = 400/503 = 0.7952\)
\(\rho/1000 = 0.7952^2 = 0.6324\)
\(\rho = 632.4\ \Omega\text{m}\).

Rounded to the nearest integer, \(\rho \approx 632\ \Omega\)m, which lies inside the accepted band of \(\boxed{630\ \text{to}\ 640\ \Omega\text{m}}\).