Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 1 / a / Solution

In cylindrical coordinates the orthogonal velocity components are $\dot{r}$, $r\dot{\theta }$, $\dot{z}$. Their squared magnitudes add, and $\dot{\mathbf{r}}\cdot \mathbf{A}=\dot{z}A(r)$. Thus $\boxed{\mathcal{L}=m({\dot{r}}^2+r^2{\dot{\theta }}^2+{\dot{z}}^2)/2+q\dot{z}A(r)}$. Taking the curl gives $B_{\theta }=-A^{\prime }(r)$, consistent with the prescribed field.