Question

Two bar magnets of same length and breadth having magnetic moments $M$ and $4M$ respectively are joined with likes poles together and suspended in a uniform magnetic field $B$, it oscillates with time period $T=\sqrt{3}\text{s}$. If the polarity of one of the magnet is reversed and the combination is again made to oscillate in the same field. Find the new time period of this combination.

• A. $\sqrt{3}\text{s}$
• B. $\sqrt{5}\text{s}$
• C. $\sqrt{2}\text{s}$
• D. $2\text{s}$

Answer 1

The correct option is B $\sqrt{5}\text{s}$

Given:
$M_1=M;M_2=4M;T_1=\sqrt{3}\text{s}$

In a combination, like poles placed together, so net dipole moment will be

$M_{res}=M_1+M_2=M+4M=5M$

And we know that, the time period of oscillation of bar magnet in uniform magnetic field is

$T=2\pi \sqrt{\frac{I}{MB}}$

So, for the combination of like poles,

$T_1=2\pi \sqrt{\frac{I}{\left(5M\right)B}}.......\left(1\right)$

In a combination, unlike poles are together, resultant moment will be

$M_{res}^{\prime }=|M_1-M_2|=|M-4M|=3M$

So time period for this combination will be

$T_2=2\pi \sqrt{\frac{I}{\left(3M\right)B}}........\left(2\right)$

On dividing eq. $\left(2\right)$ by $\left(1\right)$, we get

$\frac{T_2}{T_1}=\frac{\sqrt{5}}{\sqrt{3}}$

$T_2=\frac{\sqrt{5}}{\sqrt{3}}\times T_1=\sqrt{5}\text{s}$

Hence, option $\left(B\right)$ is correct.