Torque ${T_1}$ and ${T_2}$ are required for magnetic needles to remain perpendicular to the magnetic fields ${B_1}$ and ${B_2}$ at two different places. The ratio $\dfrac{{{B_1}}}{{{B_2}}}$is
(A) $\dfrac{{{T_2}}}{{{T_1}}}$
(B) $\dfrac{{{T_1}}}{{{T_2}}}$
(C) $\left( {\dfrac{{{T_1} + {T_2}}}{{{T_1} - {{\rm T}_2}}}} \right)$
(D) $\dfrac{{{T_1} + {T_2}}}{{{T_1} - {{\rm T}_2}}}$

Hint: A thin polarized steel pole that, when acclimated to swing in a flat plane, as in a compass, shows the course of the world's attractive fields or the inexact situation of north and south. A compass works by distinguishing the Earth's normal attractive fields. This permits the needle to all the more likely to respond to close by attractive fields.

Formula used:
Torque formula,
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{{B_1}}}{{{B_2}}}$
$T = rF\sin \theta $
$T = $ Torque
$r = $ Radius
$F = $ Force
$\theta = $ Angle between $F$ and the lever arm

Complete step by step solution:
Let ${T_1}$ and ${T_2}$ are torques ${B_1}$ and ${B_2}$ are magnetic fields perpendicular to the magnetic fields $\left( {{{\sin }^{}}{{90}^ \circ }} \right)$
$M$ be the magnetic needle
$\overline T = M \times B$
$\overline T = MB\sin \theta $$\left( {\therefore \theta = {{90}^ \circ }} \right)$
Substituting the trigonometry value,
$\overline T = {\rm M}{\rm B}\left( 1 \right)$$\left( {\therefore \sin {{90}^ \circ } = 1} \right)$
$\overline T = {\rm M}{\rm B}$
Then we hence,
${{\rm T}_1} = {\rm M}{{\rm B}_1}$
${{\rm T}_2} = {\rm M}{{\rm B}_2}$
Hence,
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{M{B_1}}}{{M{B_2}}}$
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{\not M{B_1}}}{{\not M{B_2}}}$
Canceling the similar terms we get,
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{{B_1}}}{{{B_2}}}$
Hence, the ratio of $\dfrac{{{B_1}}}{{{B_2}}} = \dfrac{{{T_1}}}{{{T_2}}}$

$\therefore $ The correct option is option (B).

Note: In material science and mechanics, force is what might be compared to straight power. It is likewise alluded to as the occasion, snapshot of power, rotational power, or turning impact, contingent upon the field of study. The idea began with the examinations by Archimedes of the utilization of switches. Attractive compasses comprise a charged needle that is permitted to pivot so it lines up with the Earth's attractive field. The finishes highlight what are known as attractive north and attractive south. Researchers and antiquarians don't have a clue when the standards behind attractive compasses were discovered. These substances are ferromagnetic.