For the life of me, I cannot remember how to solve equations similar to the cross product equations in Maxwell's equations. I haven't used vector calculus of this level in quite some time and could use some help with it. Given a specific electric field inbetween two perfect conductors (the inner conductor has an outer radius $a$ and the outer conductor has an inner radius $b$, \begin{equation} \mathbf{E}(\mathbf{r},t) = \dfrac{E_0}{2\pi} \dfrac{\hat{\mathbf{x}} x +\hat{\mathbf{y}}y}{x^2 + y^2} \cos\left(\omega\left(\frac{z}{c}-t\right)\right), \end{equation}
how would you use the equation $\nabla \times \mathbf{E} = -\frac{1}{c} \partial_t \mathbf{B}$ to solve for the magnetic field? (If there is an easier alternative way to do this, I'm open to learning that method, as well.)
Note: The notation $\partial_t$ is simply shorthand for $\partial/\partial t$.
EDIT: The coaxial cable is along the $z$-axis.
Thanks!