Understanding Biot-Savart's Law for a Rotating Disc with Current Density

[Fips]

Homework Statement

There's a disc with a current density surface σ, with radius R and angular velocity ω

Find the B

Homework Equations

α = vσ=ωrσ

The Attempt at a Solution

I'm having problems understanding why the cross product gives αr

 

[Charles Link]

$\vec{u_r}$ is a unit vector. The cross product $\vec{\alpha} \times \vec{u_r}$ is simply $\alpha \hat{z}$. Write out the expression for $dS'$ in terms of $dr$ and $d \theta$ and I think you will see where the "r" in the numerator comes from. Also, express $\alpha$ in terms of $\sigma$ and $r$ and $\omega$ . ($\alpha$ is not a constant). $\\$ editing... Your diagram is hard to read, but I see the disc extends out to radius R. (I initially thought the disc goes from $R_1$ to $R_2$.) $\\$ Once you get these steps in place, you can then evaluate the integrals.