Angular momentum in ElectroMagnetic fields(Feynman's Disk Paradox)

[Henriamaa]

In Griffiths book, "Introduction to Electrodynamics" example 8.4 he calculates the angular momentum density for a set up that is a version of Feynman disk paradox. His answer for the angular momentum points in the z direction. But if we you assume that the r vector has component in the s direction and z direction(I am almost sure this is correct) $\vec{r}$ = s$\hat{s}$+ z$\hat{z}$, then the angular momentum density has both a z component and s component. The s component is not constant. The total angular moment on the other hand has to end up with only z component or the cylinders would tip over. Where is the error in my reasoning?

 

[MisterX]

It seems to me you've made no error. The angular momentum density should in fact have an $\hat{s}$ component for $z \neq 0$. It seems Griffths neglected this. However, there is no xy component of the total angular momentum of EM field; it cancels out in via integration.