Two short magnets with their axes horizontal and perpendicular to the magnetic meridian are placed with their centres $40cm$ east and $50cm$ west of the magnetic needle. If the needle remains undeflected, the ratio of their magnetic moments ${M_1}:{M_2}$ is
(A) $2:\sqrt 5 $
(B) $64:125$
(C) $16:25$
(D) $4:5$

Hint:
In order to solve this question, we will equate the magnetic field due to both bar magnets which are opposite in direction as compared to the meridian point as needle remains undeflected and then we will solve for the ratio of magnetic moments of both bar magnets.


Formula used:
The magnetic field due to a bar magnet having magnetic moment M and at a distance of r from its centre is calculated using $B = \dfrac{{{\mu _o}2M}}{{4\pi {r^3}}}$




Complete step by step solution:
According to the question, we have given that the needle remains undeflected which means net magnetic field at the needle point is zero and since both magnets are placed at east and west which are opposite in direction so both magnetic fields magnitude must be equal in order to produce zero magnetic effect at the point. So, magnetic field due to both bar magnet can be calculated using $B = \dfrac{{{\mu _o}2M}}{{4\pi {r^3}}}$ we get,
$\dfrac{{{\mu _o}2{M_1}}}{{4\pi {{(40)}^3}}} = \dfrac{{{\mu _o}2{M_2}}}{{4\pi {{(50)}^3}}}$
on solving for the ratio ${M_1}:{M_2}$ we get
$
  \dfrac{{{M_1}}}{{{M_2}}} = {\left( {\dfrac{{40}}{{50}}} \right)^3} \\
  {M_1}:{M_2} = 64:125 \\
 $
Hence, the correct answer is option (B) $64:125$


Therefore, the correct option is B.




Note:
It should be remembered that the magnetic field produced by both magnets are opposite in direction. Magnetic meridian is a line on the earth's surface that approximates a great circle and passes through the magnetic north and south poles.