91影视

The magnetic force mainly arises amongst the particles having electrically charged and in also motion. However, the magnetic force is mainly the repulsion or attraction amongst the charged particles.

The equation of the surface element鈥檚 force is expressed as:

$B=\frac{1}{2}(B_{in}+B_{out})$

Here, is the surface element鈥檚 force, $B_{in}$and $B_{out}$are force inside and outside of the spinning shell.

The equation of the magnetic scalar potential鈥檚 curl of the inside of the shell is expressed as:

$A_{out}=\frac{{渭}_0{R}^4蝇未}{3}\frac{\sin 胃}{r^2}\widehat{鈭�}$

Here, $A_{in}$is the magnetic scalar potential鈥檚 curl of the inside of the shell, $R$is the radius, ${渭}_0$is the permeability constant, $蝇$is the angular velocity, $未$is the variability measurement, is the distance from the field, is the angle subtended and localid="1657519378591" $\widehat{蠒}$is the cap of the angle.

The cross product of $鈭�$with the $A_{in}$is expressed as:

$$\begin{gathered} 鈭�脳A_{in}=B_{in} \\ =\frac{1}{r\sin 胃}\frac{鈭�}{鈭�胃}\left(\frac{{渭}_{0}R蝇未}{3}r{\sin }^2胃\right)\hat{r}-\frac{1}{r}\frac{鈭�}{鈭�胃}\left(r\frac{{渭}_{0}R蝇未}{3}r{\sin }^2胃\right)\widehat{胃} \end{gathered}$$

The cross product of $鈭�$with the $A_{out}$is expressed as:

$$\begin{gathered} 鈭�脳A_{out}=B_{out} \\ =\frac{1}{r\sin 胃}\frac{鈭�}{鈭�胃}\left(\frac{{渭}_0{R}^4蝇未}{3r^2}{\sin }^2胃\right)\hat{r}-\frac{1}{r}\frac{鈭�}{鈭�胃}\left(r\frac{{渭}_0{R}^4蝇未}{3r^2}{\sin }^2胃\right)\widehat{胃} \\ =\frac{2}{3}{渭}_0{R}^4未蝇\frac{1}{r^3}(2\cos 胃\hat{r}+\sin 胃\widehat{胃}) \end{gathered}$$

The equation of the average surface of the sphere is expressed as:

$B_{avg}=\frac{1}{2}{\left[(B_{in}+B_{out})\right]}_{r-R}$

The above equation can be reduced as:

$$\begin{gathered} B_{avg}=\frac{1}{2}\left[\frac{1}{3}{渭}_0{R}^4未蝇\frac{1}{R^3}(2\cos 胃\hat{r}+\sin 胃\widehat{胃})+\frac{2}{3}{渭}_{0}R未蝇(\cos 胃\hat{r}-\sin 胃\widehat{胃}\right] \\ =\frac{1}{6}R蝇未渭(4\cos 胃\hat{r}-\sin 胃\widehat{胃}) \end{gathered}$$

The equation of the force on the differential surface element is expressed as:

$$\begin{gathered} dF=dqv脳B_{avg} \\ =未dSv脳B_{avg} \end{gathered}$$ 鈥�(颈)

Here, dqv is the differential surface element.

The equation of the velocity of the shell is expressed as:

$v=R\sin 胃蝇\widehat{蠒}$

The differentiation of the above equation is expressed as:

$dS=R^2\sin 胃d蠒d胃$

Substituting $R\sin 胃蝇\widehat{蠒}$for $v$and $R^2\sin 胃d蠒d胃$for $dS$in the equation (i).

$$\begin{gathered} dF=\frac{1}{6}{渭}_0{未}^2{蝇}^2{R}^4{\sin }^2胃d胃d蠒\widehat{蠒}脳(4\cos 胃\hat{r}-\sin 胃\widehat{胃}) \\ =\frac{1}{6}{渭}_0{未}^2{蝇}^2{R}^4{\sin }^2胃(4\cos 胃\widehat{胃}-\sin 胃\hat{r})d胃d蠒 \end{gathered}$$

The equation of the z component of the force of the shell is expressed as:

$d{F}_z=dF鈥�\hat{z}$

鈥�(颈颈)

Here, $dF$is the differential force and $\hat{z}$is the position vector.

The equation of the angle subtended by the force is expressed as:

$\widehat{胃}=\cos 胃\hat{s}-\sin 胃\hat{z}$

The equation of the position vector in the $z$direction is expressed as:

localid="1658556502484" $\hat{\mathit{r}}=\mathbf{sin}胃\hat{s}+\mathbf{cos}胃\hat{\mathbf{z}}$

Substituting the values in the equation (ii).

$$\begin{gathered} dF=\frac{1}{6}{渭}_0{蝇}^2{未}^2{R}^4{\sin }^2(4\cos 胃\sin 胃-\cos 胃\sin 胃)d胃蠒 \\ =\frac{1}{2}{渭}_0{蝇}^2{未}^2{R}^4{\sin }^3胃\cos 胃d胃d蠒 \end{gathered}$$

Double integrating the above equation with the integral limits can be expressed as:

$$\begin{gathered} F=\frac{1}{2}{渭}_0{未}^2{蝇}^2{R}^4{鈭�}_0^{2蟺}{鈭�}_0^{蟺/2}{\sin }^3胃\cos d胃d蠒 \\ =蟺{渭}_0{蝇}^2{未}^2{R}^4{鈭�}_0^{\mathrm{蟺}/2}{\sin }^3\mathrm{胃肠辞蝉}\mathrm{诲胃} \end{gathered}$$

鈥�(颈颈颈)

The change in the variables can be expressed as:

$$\begin{gathered} x=\sin 胃 \\ dx=\cos 胃d胃 \end{gathered}$$

Substitute the value of the integral in the equation (iii).

$$\begin{gathered} F=蟺{渭}_0{蝇}^2{未}^2{R}^4{鈭�}_0^1{x}^{3}dx \\ =蟺{渭}_0{蝇}^2{未}^2{R}^4\left(\frac{{\mathrm{x}}^4}{4}\right) \\ =蟺{渭}_0{蝇}^2{未}^2{R}^4\left(\frac{1}{4}\right) \\ =\frac{蟺}{4}蟺{渭}_0{蝇}^2{未}^2{R}^4 \end{gathered}$$

Thus, the magnetic force of attraction is $\frac{\mathrm{蟺}}{4}{渭}_0{蝇}^2{未}^2{R}^4$.