Question

Assume that the terrestrial magnetism is due to a short magnet at the centre of the earth. Refer to a point on the surface of the earth. Given : magnetic latitude of the reference point = $\lambda ,\delta$ = angle of dip and R = radius of earth. Then, the value of vertical component V of the earth's field is

• A. $\frac{2Msin\lambda }{R^3}$
• B. $tan\delta$
• C. $tan\lambda$
• D. $2tan\lambda$

Answer 1

Answer: (D)

$2tan\lambda$

$\text{Solution}:$

Considering the situation for dipole, at position $(r,\theta )$ we have,

$B_r=\frac{{\mu }_0}{4\pi }\frac{2Mcos\theta }{r^3}$ and $B_{\theta }=\frac{{\mu }_0}{4\pi }\frac{Msin\theta }{r^3}$

And as $tan\varphi =\frac{B_V}{B_H}=-\frac{B_r}{B_{\theta }}$ ,

so in the light of Eq. (1)

$tan\varphi =-2cot\theta$,

But from firgure

$\theta ={90}^o+\lambda$

So,

$tan\varphi =-2cot({90}^o+\lambda )$

,i.e, $tan\varphi =2tan\lambda$

$\text{Hence D is the correct option}$