Question

Two short magnets have equal pole strengths but one is twice as long as the other. The shorter magnet is placed 20 cm in tan A position from the compass needle. The longer magnet must be placed on the other side of the magnetometer for no deflection at a distance equal to

• A. 20 cm
• B. $20\times (2{)}^{4/3}cm$
• C. $20\times (2{)}^{2/3}cm$
• D. $20\times (2{)}^{1/3}cm$

Answer 1

The correct option is D $20\times (2{)}^{1/3}cm$

In tan A position of the magnetometer, the compass needle is in end-on (axial) position of bar magnet.

Let two magnets are placed in position (1) and (2) as shown, P = position of compass needle



Let m= pole strength of each magnet

Magnetic moment of magnet 1,

$M_1=ml$

Magnetic moment of magnet 2,

$M_2=m\left(2l\right)=2M_1$

To get zero deflection, net magnetic field at point P should be zero. So, both magnets should produce equal and opposite field at P

Taking field at axial position P of both magnets we get,

$B_1=B_2$

$\Rightarrow \frac{{\mu }_0}{4\pi }\times \frac{2M_1}{r_1^3}=\frac{{\mu }_0}{4\pi }\times \frac{2M_2}{r_2^3}$

$\Rightarrow \frac{M_1}{M_2}={\left(\frac{r_1}{r_2}\right)}^3$

$\Rightarrow \frac{1}{2}={\left(\frac{20cm}{r_2}\right)}^3$

$\therefore r_2=20\times (2{)}^{1/3cm}$

Hence, the correct option is (B)