# Electric Field in an Infinite Alternating Current Solenoid

Reading Wangsness's Magnetic Fields book I stumbled upon an explanation I can't understand. It goes like this: There's an alternating current solenoid with radius $a$. We know the magnetic induction is

$\textbf{B} = B_{0}cos(\omega t+\alpha)\textbf{k}$ for $r\leq a$, and zero for $r\geq a$.

It says that due to the problem's symmetry we can expect an electric field on the xy-plane with just a $\phi$ component so

$\displaystyle\oint_C\textbf{E}\cdot d\textbf{s}=\oint_C E_{\phi}\rho d\phi=2\pi\rho E_{\phi}$.

My question is: How de we know we can expect this? Why not to expect a $\rho$ component for the field?