Question

hi
we have a disk with uniform charge density "sigma" that rotate around of axis that transmitted from it's origin,find magnetic moment of disk?

Answer 1

Ok,

Total charge on the disk is: \[\ Q=\sigma \pi R^2 \]where R is the radius of the disk

If you divide the disk into small rings(width dr) you can write the current in that ring as:\[\ dI=\frac{dQ}{T}\]where T- time period
Since \[\ dI =2 \sigma \pi rdr \]
and \[\ T=\frac{2\pi}{\omega}\]
you can rewrite dI as:\[\ dI =2 \sigma \pi rdr \frac{\omega}{2 \pi} \]\[\ dI = \sigma r \omega dr \]

Magnetic moment is defined as: \[\ m=IA\] where I-current, A-area
\[\ dm=dI \pi r^2\]\[\ dm= \sigma r \omega dr \pi r^2\]\[\ \int dm= \int \sigma \omega \pi r^3 dr\]\[\ m=\frac{\pi}{4}\sigma \omega R^4\]

That would be it.

Answer 2

Also, put a hat on m and omega, since it is a vector...

Answer 3

thnx