Question

Assume the dipole model for earth's magnetic field B which is given by thevertical component of magnetic field, $B_V=\frac{{\mu }_o}{4\pi }\frac{2m\cos \theta }{r^3}$ and the horizontal component of magnetic field $B_H=\frac{{\mu }_o}{4\pi }\frac{m\sin \theta }{r^3}$, where $\theta ={90}^o-$latitude as measured from magnetic equator, then the loci of point for which dip angle is $\pm {45}^o$.

• A. ${\tan }^{-1}\left(3\right)$
• B. ${\tan }^{-1}\left(2\right)$
• C. ${\tan }^{-1}\left(0.5\right)$
• D. ${\tan }^{-1}\left(1\right)$

Answer 1

The correct option is B ${\tan }^{-1}\left(2\right)$

Here, $B_V=\frac{{\mu }_o}{4\pi }\frac{2m\cos \theta }{r^3}$

$B_H=\frac{{\mu }_o}{4\pi }\frac{m\sin \theta }{r^3}$ and $\delta ={45}^o$

$\because \tan \delta =\frac{B_V}{B_H}$

Hence, $\tan {45}^o=\frac{2\cos \theta }{\sin \theta }=2\cot \theta$

$\therefore 1=\frac{2}{\tan \theta }$ $\Rightarrow \tan \theta =2$

or $\theta ={\tan }^{-1}\left(2\right)$ is the loci of points.