The law of Biot and Savart generalizes to the case of surface currents as
$\vec{B} = \frac{\mu_0}{4\pi} \int \frac{\rho \vec{v} \times d\vec{a}}{r^2}$
where $\rho$ is the local charge density, $\vec{v}$ is the local velocity, and $d\vec{a}$ is a differential area element.
A cylindrical shell with radius R and length W carries a uniform charge Q and rotates about its axis with angular speed $\omega$. The center of the cylinder lies at the origin O and its axis is coincident with the z-axis, as shown in (Figure 1).
Use the above equation as an alternative means to derive the magnetic field at the center of the cylinder.
Part C
What is the velocity $\vec{v}$ of the element?
Express your answer in terms of the variables z, R, $\phi$, and $\omega$, if needed. Enter the z, y, and z components of the velocity separated by commas.
-$\omega$R sin $\phi$,$\omega$R cos $\phi$,0
Part D
What is its vector product by the unit vector $\hat{r}$, $\vec{v} \times \hat{r}$?
Express your answer in terms of the variables z, R, $\phi$, and $\omega$, if needed. Enter the x, y, and z components of the vector product separated by commas.
