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

CSIR - NET DEC 2014

107) Electric measurements are made over a homogeneous medium passing current through widely separated Current electrodes and placing the potential electrodes at distance of 10m and 20m from one of the current electrodes. If the resistance measured is 2Ω, the resistivity of the medium is

A) 20πΩm 

B) 40 πΩm 

C) 60 πΩm 

D) 80 πΩm

Solution:

Based on the above data it is pole-dipole configuration

$\rho _a = 2\pi \frac{ab}{b-a} \frac{\triangle V}{I} ----- (1)$

a = 10m

b = 20m

if b=2a then,

$\rho_a=2\pi \frac{2a^2}{2a-a} \frac{\triangle V}{I}=2\pi \frac{2a^2}{a} \frac{\triangle V}{I}=4\pi a (\frac{\triangle V}{I})-------(2)$

$Resistance(R)=\frac{\triangle V}{I}=2Ω$

$\rho_a=4\times \pi \times 10 \times 2=80\pi Ωm$

$\therefore apparent resistivity, \rho_a=80\pi Ωm$