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

CSIR – NET JUNE 2013

88 b) A two-dimensional body of susceptibility K striking N30^oE at the magnetic latitude 30^oN is magnetised by induction in the Earth’s magnetic field F. The effective intensity and inclination of the induced magnetism are respectively

1) KF, 30^O

$2)\frac{KF}{2},\tan^{-1}(\frac{2}{\sqrt{3}})$

$3) (\sqrt{\frac{19}{28}})KF,\tan ^{-1}(4/\sqrt{3})$

$4) (\sqrt{\frac{19}{7}})KF,\tan ^{-1}(\sqrt{3}/4)$

Solution:

Formula,  

 $1)J=KF \sqrt{1-\cos^2\alpha.\cos^2i} ---(1)$

J – Intensity of effective magnetization of 2-D body, lying in a plane perpendicular to the strike

α – strike of the body=N30^oE

i = dip of the Earth’s magnetic field 
 tan i=2tan φ
Latitude is given that  φ= 30 degrees

$ tan i=2tan30^o$

$ tan i=(2\times\frac{1}{\sqrt{3}})$
 

$Dip (or) inclination( i)=\tan^{-1}(\frac{2}{\sqrt{3}})=49^o$

K= Susceptibility

F= earth magnetic field

Substitute in above values in eq.(1)

$J=KF\sqrt{1-  \cos^2(30^o).\cos^2(49^o)}$

$J=KF\sqrt{1-(0.75).(0.43)}$

$J=KF\sqrt{1-0.322}$

$J=KF\sqrt{0.678}$

$J=KF\sqrt{\frac{19}{28}}$

$J=(\sqrt{\frac{19}{28}} )KF$

2. Effective inclination (or) dip(i_eff)

(i_eff – dip of the effective magnetization)

$i_{eff}=\tan^{-1}(\frac{tan i}{sin\alpha})$

$i – dip of the earth’s magnetic field=49^o$

$α – strike of the body – N30^oE$

$i_{eff}=\tan^{-1}(\frac{tan 49^o}{sin30^o})$

$i_{eff}=\tan^{-1}(1.15\times2)$

$i_{eff}=\tan^{-1}(2.3)$

$i_{eff}=\tan^{-1}(\frac{4}{\sqrt{3}})$

Reference: I V Radhakrishna Murthy,  Magnetic anomalies of two-dimensional bodies and algorithms for magnetic inversion of dykes and basement topographies, Proc.Indian Acad. Sci(Earth Planet.Sci),Vol 99, no 4, Dec 1990 , pp, 549-579.