Csir_Net_Dec_2017
112) what
is the value of the Earth’s Magnetic field (in Oe) at a place where it’s
vertical and horizontal components are equal if the magnetic field at the
equator is 0.30 Oe.
(Special thanks to Y.Suresh sir, GSI)
(Thanks to Chandrasekhar, ANU)
Solution:
Given that Magnetic field at the equator = 0.30 Oe
Find the Magnetic field at a place where its B_{V} =B_{H}
We know that Earth’s total magnetic field
B_{F}=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\cos^{2}\theta+1}-------------------(1)
Where m= magnetic moment
R= radius
\theta =
co-latitude
The Earth’s total
magnetic field in terms of geographical latitude (\phi) is as below
B_{F}=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\sin
^{2}\phi +1}
As per given data at
the equator the total magnetic field is 0.30 Oe means at a geographical latitude
00
i.e
B_{F}=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\sin
^{2}\phi +1}
B_{F(equatotr}=0.30
\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\sin ^{2}\phi
+1}=0.30
\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\sin ^{2}(0)
+1}=0.30
\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{0 +1}=0.30
\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}=0.30
m=\frac{0.30
\times4\pi r^{3}}{\mu_{0}}--------------------------(2)
Now as per given assume that B_{Z} =B_{H}
We know that the radial and the tangential components are
as below
B_{r}=B_{Z}=\frac{\mu_{0}}{4\pi}\frac{2m\cos\theta}{r^{3}}
And
B_{\theta}=B_{H}=\frac{\mu_{0}}{4\pi}\frac{m\sin\theta}{r^{3}}
Therefore
B_{Z} =B_{H}
\frac{\mu_{0}}{4\pi}\frac{2m\cos\theta}{r^{3}}=\frac{\mu_{0}}{4\pi}\frac{m\sin\theta}{r^{3}}
2\cos\theta=\sin\theta
\frac{\cos\theta}{\sin\theta}=\frac{1}{2}
\cot\theta=0.5
\theta=\cot^{-1}(0.5)
\theta=63.43^{0}
Now value of the
magnetic field at B_{V} =B_{H} is
B_{F}=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\cos^{2}\theta+1}
By substituting the m and \theta values in the above
equation we got
B_{F}=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{3\cos^{2}\theta+1}
B_{F}=(\frac{\mu_{0}}{4\pi r^{3}}){(\frac{0.30 \times4\pi r^{3}}{\mu_{0}})}{}\sqrt{3\cos^{2}(63.43^{0})+1}
B_{F}=0.30\sqrt{3\cos^{2}(63.43^{0})+1}
B_{F}=(0.30)(\sqrt{1.6})
B_{F}=0.379
Oe
Answer would be 0.38Oe.
ReplyDeleteOk.. i will check it once...
DeleteEquate horizontal and vertical components of field. Find magnetic latitude. Find expression for M using horizontal compenent. Put the value of M (The expression including constants and r) in vertical field equation. Now we get vertical compenent it's about 0.268 Oe. Find the total field by multiplying it with √2 (Since total field is the vector sum of same vertical and horizontal compenents) ans will be 0.38Oe.
ReplyDeleteCan you please send the solution if possible?
Delete