Two short bar magnets A and B are arranged co – axially. The distance between their centres is 30cm. A compass needle placed on their axis at a distance of 6cm from B shows no deflection. The ratio of the magnetic moments of A and B is (Neglect earth’s magnetic field)

Given,

distance between the magnets, d = 30cm

dompass needle distance from magnet, B = 6cm

Now, distance of magnet A from needle,

${\text{d}}_1=30-6=24\text{cm}$

and distance of magnet B from needle, ${\text{d}}_{\text{1}}=\text{6}\text{cm}$

$\therefore$  Magnetic field on a axial line is given as, $\text{B}=\frac{{\mu }_{\text{o}}}{4\pi }\frac{\text{2M}}{{\text{d}}^{\text{1}}}$

Hence, the ratio of the magnetic moment for A and B

$\frac{{\text{M}}_{\text{1}}}{{\text{d}}_{\text{1}}^{\text{1}}}=\frac{{\text{M}}_{\text{2}}}{{\text{d}}_{\text{2}}^{\text{1}}}\Rightarrow \frac{{\text{M}}_{\text{1}}}{{\text{M}}_{\text{2}}}={\left(\frac{{\text{d}}_{\text{1}}}{{\text{d}}_{\text{2}}}\right)}^3$

$\therefore {\text{d}}_1=24\text{cm},{\text{d}}_2=6\text{cm}$

$\frac{{\text{M}}_{\text{1}}}{{\text{M}}_{\text{2}}}={\left(\frac{24}{6}\right)}^3$

$\frac{{\text{M}}_{\text{1}}}{{\text{M}}_{\text{2}}}={\left(4\right)}^3=64$

So, the ratio of the magnetic moments of A and B is 64 : 1