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

113. A potential difference of 10 mV is measured in an electrical survey with Wenner configuration using an electrode spacing of 10 m and 1.0 amp current. What would be the potential difference if the measurements are made with a Schlumberger configuration employing current electrode spacing of AB =30 m, potential electrode spacing of MN = 4 m and the same current?

1. 27.6 mV

2. 2.76 mV

3. 3.62 mV

4. 36.2 mV

In Wenner configuration:

Apparent resistivity (ρ_a ) = 2 π a (ΔV/ I) ----------------(1)

First calculate the apparent resistivity, then using this calculate the potential difference (ΔV) in schlumberger configuration.

 

Electrode spacing a= 10 m

(Note : If in any wenner configuration problems current electrode spacing is given, that should be divided by 3.)

Current = 1 amp

Potential difference (ΔV) =10 mV

Substitute these values in equation (1)

ρ_a = 2 * 3.14 * 10 *( 10/ 1)

ρ_a = 628 Ω m

In Schlumberger configuration:

Apparent resistivity (ρ_a ) = π /4 * (L^{^2} -l^² ) / l * (ΔV / I ) -------------(2)

we know the apparent resistivity (ρ_a ) = 628 Ω m

Current electrode spacing (AB) or (L)= 30 m

Potential electrode spacing (MN) or (l) = 4 m

ΔV = (ρ_a * 4 * l * I ) / (L^² – l^² ) * π

ΔV = (628 * 4 * 4* 1 ) / (30^{^2} – 4^{^2} ) * 3.14

 

ΔV = 3.619 mV