Question

A magnetic wire of dipole moment 4$\pi A{{m}^{2}}$ is bent in the form of a semicircle. The new magnetic moment is?

(A) 4$\pi A{{m}^{2}}$

(B) 8$\pi A{{m}^{2}}$

(C) 4$A{{m}^{2}}$

(D) 8$A{{m}^{2}}$

Answer 1

Magnetic dipole is the analogous form of electric dipole. The dipole moment is given by the magnetic moment multiplied with the separation between the two dipoles. So, with this formula, we can calculate the magnetic moment for the semi-circular wire. The important point to remember here is that the magnetic moment is dependent on the distance between the two dipoles.

Formula used:

The magnetic moment is given by,

$M=m{l_m}$

where,

m- The magnetic strength

${l_m}$- The magnetic length

Complete step-by-step solution:

Given,

Magnetic dipole moment of the straight wire=M=4$\pi A{{m}^{2}}$

Let the initial length of the wire be $L$. Then, it is given that the wire is bent in the form of a semicircle. Then the length of the wire changes to,

$L= \pi {R}$

$\Rightarrow R= \dfrac {L}{\pi}$

Now, we know that the magnetic length is the nearest distance between the poles. Therefore we get the diameter of the semicircle formed as the magnetic length. Thus we have:

$\Rightarrow {l_m}= \;diameter\;of \;the\; semicircle= 2R$

$\Rightarrow {l_m}= 2\times \dfrac {L}{\pi}$

Thus, ${l_m}$ changes to $\dfrac{2L}{\pi }$. $\therefore$ The new magnetic moment,

${M}'=m\times \dfrac{2L}{\pi }$

$\Rightarrow {M}'=\dfrac {2}{\pi} \left ( m L \right )$

$\Rightarrow {M}'=\dfrac {2}{\pi} M$

$\Rightarrow {M}'=\dfrac {2}{\pi} \times 4{\pi}$

$\Rightarrow {M}'= 8 A {m^2}$

(Here we took M=mL, because it is the initial magnetic moment of the wire)

Thus, the magnetic moment turns out as 8$A{{m}^{2}}$.

So, the correct answer is “Option D”.

Additional information:

A dipole is used to represent a field created by a complex system of charges. Electric, magnetic and current dipoles are the 3 main types of dipoles available as systems. Magnetic dipole moment or magnetic moment is used to characterize magnetic properties of matter. It is a useful concept even in case of atoms and molecules where the charge separation is measurable and charges are minutely small. Infact, most elementary charges behave intrinsically as magnetic dipoles.

Note: In this problem, the magnetic dipole is seen to change with the configuration of the wire from a straight wire to that of a bent circular wire. The two main quantities on which dipole moment is dependent is on the magnetic moment of the system and the separation distance between the dipoles. The important point to remember here is that the magnetic moment is dependent on the distance between the two dipoles.