Question: Q. 2. (a) Write the expression for the equivalent magnetic moment of a planer current loop of area $A$, having $N$ turns and carrying a current $i$. Use the expression to find the magnetic dipole moment of a revolving electron.
(b) A circular loop of radius $r$, having $N$ turns and carrying current $I$, is kept in the $X Y$ plane. It is then subjected to a uniform magnetic field $B=B_{x} \hat{I}+$ $B_{y} \hat{J}+B_{z} \hat{k}$. Obtain expression for the magnetic potential energy of the coil-magnetic field system.
R&A [CBSE SQP 2018-19]
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Solution:
Ans. (a) The equivalent magnetic moment is given by $\quad \mu=\mathrm{NiA} \quad 1 / 2$ The direction of $\mu$ is perpendicular to the plane of current carrying loop. It is directed along the direction of advance of a right-handed screw rotated along the direction of flow of current. $1 / 2$ Derivation of expression for $\mu$ of electron revolving around a nucleus
(b) for the loop,
$$ \vec{\mu}=\mathrm{N}\left(\mu r^{2}\right) i( \pm \hat{k}) \quad 1 / 2 $$
Magnetic potential energy $=\vec{\mu} \cdot \vec{B}$
$=N\left(\mu r^{2}\right) i( \pm \hat{k}) \cdot\left(B_{x} \hat{i}+B_{y} \hat{j}+B_{z} \hat{k}\right)$
$= \pm r^{2} N I B_{z}$.
$1 / 2$
[CBSE Marking Scheme, 2018-19]
Detailed Answer :
(a) The equivalent magnetic moment is given by $\mu=N i A$ The direction of $\mu$ is perpendicular to the plane of current carrying loop. It in directed along the direction of advance of a right handed screw rotated along the direction of flow of current. $1 / 2$ Derivation :
Let us consider an atom with an electron revolving around the nucleus.
Charge on electron is $e$. Let its mass be $m$.
Radius of electron’s orbit is $r$.
Area of electron’s orbit is $A=\pi r^{2}$
Magnetic dipole moment of the atom is,
$$ \mu=i A N $$
The current $i$ is given by,
$$ i=\frac{e}{T} $$
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