Question

The magnetic field at the centre of a circular loop of area A is B. The magnetic moment of the loop is
A. $\dfrac { B{ A }^{ 2 } }{ { \mu }_{ 0 }\pi }$
B. $\dfrac { B{ A }\sqrt { A } }{ { \mu } }$
C. $\dfrac { B{ A }\sqrt { A } }{ { { \mu }_{ 0 } }\pi }$
D. $\dfrac { 2B{ A }\sqrt { A } }{ { { \mu }_{ 0 } }\sqrt { \pi } }$

Answer 1

Hint: The magnetic field of the loop is given; we should substitute the area of the loop in the magnetic intensity formula to gain the magnetic moment. The equation for magnetic field can be written as $B=\dfrac { { \mu }_{ 0 }I }{ 2r }$. Where, I is the magnetic intensity and its equation is given as $I=\dfrac { m }{ A }$.

Complete step by step answer:
A magnetic field is the area or space where a magnetic pole experiences a force or due to the space around it. Magnetic induction is the one of the fundamental characteristics of the magnetic fields.
The given magnetic field of the loop is ‘B’. Let us assume the radius of the loop to be ‘r’.
Then, the area of the circular loop is given by, A = πr2 or $r=\sqrt { \dfrac { A }{ \pi } }$
Substituting the radius of loop ‘r’ in the magnetic field formula, we get
$B=\dfrac { { \mu }_{ 0 }I }{ 2r }$
$B=\dfrac { { \mu }_{ 0 }I }{ 2\sqrt { \dfrac { A }{ \pi } } }$
From which, the intensity of magnetization is derived,
$I=\dfrac { 2B }{ { \mu }_{ 0 } } \sqrt { \dfrac { A }{ \pi } }$
Intensity of magnetization is the representation of extent to which a material has been magnetized under the magnetic field. It is denoted as ‘I’ and defined as the magnetic moment per unit volume of the material. The equation is given as,
$I=\dfrac { m }{ A }$ or m = IA
Substituting the value of ‘I’ in the above equation. We get,
$M=\dfrac { 2B }{ { \mu }_{ 0 } } \sqrt { \dfrac { A }{ \pi } } A$
 $M=\dfrac { 2BA }{ { \mu }_{ 0 } } \sqrt { \dfrac { A }{ \pi } }$
Therefore, the correct answer for the given question is option (D).

Note: Magnetic intensity is also defined as the pole strength per unit area of the cross section. Its unit is $\text{A}{{\text{m}}^{\text{-1}}}$ Any magnet with two poles has a magnetic dipole. Magnetic moment of a magnet is the product of pole strength and the distance between the two poles.