91影视

The time period of rotation of donut is given as:

$t=\frac{蟺}{蝇}$

The current in the donut is given as:

$\backslash begin\left\{aligned\right\}\&I=\backslash frac\left\{Q\right\}\left\{t\right\}\backslash \backslash \&I=\backslash frac\left\{Q\right\}\{\backslash left(\backslash frac\{2\backslash pi\}\{\backslash omega\}\backslash right\left)\right\}\backslash \backslash \&I=\backslash frac\{\backslash text\{I\left\}\right\}\{\backslash text\{\mathrm{啶�}\left\}\right\}\backslash end\left\{aligned\right\}$

The cross sectional area of the donut is given as:

$A=蟺R^2$

The magnetic dipole moment of donut is given as:

$m=1路A$

$\backslash begin\left\{aligned\right\}\&m=\backslash left(\backslash frac\{Q\backslash omega\left\}\right\{2\backslash pi\}\backslash right)\backslash left(\backslash piR^\{2\}\backslash right)\backslash \backslash \&m=\backslash frac\{Q\backslash omegaR^\{2\left\}\right\}\left\{2\right\}\backslash end\left\{aligned\right\}$

The angular momentum of donut is given as:

$L=M蝇R^2$

The gyromagnetic ratio of the donut is given as:

$g=\frac{m}{L}$

Substitute all the values in the above equation.

$\backslash begin\left\{aligned\right\}g\&=\backslash frac\left\{2\right\}\{M\backslash omegaR^\{2\left\}\right\}\backslash \backslash g\&=\backslash frac\left\{Q\right\}\{2M\}\backslash end\left\{aligned\right\}$

Therefore, the gyromagnetic ratio of donut is $\frac{Q}{2M}$.

The gyromagnetic ratio of the donut does not depend on the geometric feature (radius) of donut so the gyromagnetic ratio of uniform spinning sphere would be same as of donut.

Therefore, the gyromagnetic ratio of uniform spinning sphere is also$\backslash frac\left\{Q\right\}\{2M\}.$

Consider the expression for the magnetic dipole moment:

$m_e=\frac{e\sqrt{4m}}{4m}$

Here, 鞁� is Planck's constant and its value is $1.05脳{10}^{-34}\mathrm{J}路\mathrm{s}$, e is the charge of electron and its value is $1.6脳{10}^{-19}\mathrm{C},m$is the mass of electron and its value is $9.11脳{10}^{-31}\mathrm{kg}$.

Substitute all the values in the above equation.

$\backslash begin\left\{aligned\right\}\&m\_\left\{e\right\}=\backslash frac\{\backslash left(1.6\backslash times10^\{-19\}\backslash mathrm\left\{C\right\}\backslash right)\backslash left(1.05\backslash times10^\{-34\}\backslash mathrm\{~J\}\backslash cdot\backslash mathrm\left\{s\right\}\backslash right\left)\right\}\{4\backslash left(9.11\backslash times10^\{-31\}\backslash mathrm\{~kg\}\backslash right\left)\right\}\backslash \backslash \&m\_\{\backslash theta\}=4.61\backslash times10^\{-24\}\backslash mathrm\{~A\}\backslash cdot\backslash mathrm\left\{m\right\}^\left\{2\right\}\backslash end\left\{aligned\right\}$

Therefore, the magnetic dipole moment of electron is$4.61脳{10}^{-24}\mathrm{A}路{\mathrm{m}}^2$.