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

 Q) A strongly magnetic spherical specimen when exposed to a magnetic field of 18.8 Oe exhibited an induced magnetism of 2.1 gauss. The susceptibility of the specimen (in cgs units) is   

(Special thanks to Rohit Jha & Chandrasekhar, AKNU)

Solution:

In general, induced magnetism(M) is defined as the product of magnetic susceptibility(K) and the external magnetic field(H).

              $M=K\times H$----------------------------(1)

 Given that M = 2.1 gauss

                    H = 18.8 Oe.

Here we cannot directly calculate the magnetic susceptibility because there is a difference in the induced magnetic field and the external magnetic field though they have the same sign, which gives the concept of Demagnetization.

               When there is Demagnetization, we use the apparent magnetic susceptibility, K_a.

Then eq(1) becomes $M=K_{a}\times H$

$K_{a}=\frac{M}{H}$

$K_{a}=\frac{2.1}{18.8}=0.111$-------------------------(2)

Where,

$K_{a}=\frac{K}{1+NK}$-------------------------------(3)

 

For the spherical objects $N=\frac{4\pi}{3}$

Substitute in eq(3) we get

$K_{a}=\frac{K}{1+(\frac{4\pi}{3})K}$

 

$\frac{K}{1+(\frac{4\pi}{3})K}=0.111$

 

$K=0.111(1+(\frac{4\pi}{3})K)$

 

$K=0.111+0.4647\times K$

 

$K(1-0.4647)=0.111$

 

$K(0.5353)=0.111$

 

$K=\frac{0.111}{0.5353}$

 

$k=0.207\sim 0.21$

Reference:  Applied Geophysics by W.M. Telford , L.P. Geldart and R.E.Sheriff.