GATE-2018 (24) :Wyllie – Time average equation

GATE-2007

In electromagnetic method of geophysical prospecting , the depth of investigation (skin depth) is a function of the physical property of the medium and frequency of the source field.

Q78. A homogeneous medium is represented by the electrical conductivity ‘σ’ and magnetic permeability ‘μ’ . If the angular frequency of the source field is ‘ω’ , then the expression for the skin depth ‘d’ is

A) d= √((ω*μ*σ)/ 2)

B)d= √((2*σ)/ (ω*μ))

C) d= √(1/ (2*ω*μ*σ))

D) d= √(2/ (ω*μ*σ))

Solution:

Ratio of electrical and magnetic field intensity (E/H) termed as the impedence (Z) is a characterstic measure of the EM properties of the sub surface medium.

For a plane wave,

Z= E_X /E_Y ---------------------(1)

= ((i* ω*μ*)/ K)

K= √(i*ω*μ*σ)

substitute above values in eq (1)

Z= ( (i*ω*μ) / √(i*ω*μ*σ) )

impendence Z= √( (i*ω*μ)/ σ )

skin depth in a homogeneous earth d = √(2 /K)

d= √(2/ (ω*μ*σ)) -------------------------- (2)

Q79. The frequency of the EM source required to achieve a depth of investigation of 1km in a medium of electrical resistivity of 40 Ωm and magnetic permeability of 4П* 10^^-7 H/m is

A) 1 Hz       B) 10 Hz            C) 100 Hz           D) 1000 Hz

Solution:

Skin depth (d)= √(2/ (ω*μ*σ)) -------------------------- (1)

ω= 2Пf

magnetic permeability (μ)= 4П* 10^{^-7} H/m

electrical resistivity (ρ) = 4. 0 Ωm

electrical conductivity (σ) = 1/ρ = 1/ resistivity

skin depth (d) = 1 km

frequency (f) =?

above values substitute in eq (1)

1000 = √ ((2* 4.0)/ (2*П*f* 4П* 10^^-7))

f= (8 / (2* 3.14 *4 *3.14 * 10^{^-7} *10^⁶))

f= (8 / (2* 3.14 * 4*3.14 *0.1))

f= 1.015 Hz

f= 1 Hz