First slide

Magnetic dipole - Moving charges and magnetism

Question

An insulating rod of length L carries a charge ‘q’ distributed uniformly on it. The rod is pivoted at its mid point and is rotated at a frequency f about a fixed axis perpendicular to the rod and passing through the pivot. The magnetic moment of the rod system is

Moderate

A

$\frac{1}{12} π q f l^{2}$
B

$π q f l^{2}$
C

$\frac{1}{6} π q f l^{2}$
D

$\frac{1}{3} π q f l^{2}$

Solution

At a distance ‘x’ consider a small element of width ‘dx’ .
As the element rotates, it can be considered as a current carrying loop.
Magnetic moment of the small element is
$d M = I A = \frac{d q}{T} π x^{2} = \frac{q}{l} d x f π x^{2}$
$\Rightarrow M = \frac{q f π}{l} \int_{- l / 2}^{l / 2} x^{2} d x = \frac{q π f l^{2}}{12}$ .

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