Question

Assume the dipole model for earth's magnetic field B which is given by$B_V$= vertical component of magnetic field $\frac{{\mu }_0}{4\pi }=\frac{2mcosecos\theta }{r^3}$
$B_H$ Horizontal component of magnetic field $\frac{{\mu }_0}{4\pi }=\frac{sinsin\theta }{r^3}$

$\theta ={90}^{\circ }$latitude as measured from magnetic equator. Find loci of points for which $\left|B\right|$ is minimum.

Answer 1

Given, vertical component of earth's magnetic field, $B_v=\frac{{\mu }_o}{4\pi }\frac{2mcos\theta }{r^2}$

Horizontal component of earth's magnetic field$B_H=\frac{{\mu }_0}{4\pi }\frac{msin\theta }{r^3}$

$B_{net}=\sqrt{B_V^2+B_H^2}=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}\sqrt{4co{s}^2\theta +si{n}^2\theta }=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}\sqrt{3co{s}^2\theta +1}$

$\left|B_{net}\right|$ is minimum if, $cos\theta =0or\theta =\frac{\pi }{2}$

$\left|B_{net}\right|$is minimum at magnetic equator.

Final Answer: Magnetic equator.