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