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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?

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According to the Maxwell equations, in the areas where the current density is zero, the temporal derivative of the electric field equals the curl of the magnetic field (up to a constant factor). You may wish to look at the expression for a curl of a vector function in cylindrical coordinates (http://isites.harvard.edu/fs/docs/icb.topic970148.files/Spherical_coord.pdf)

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