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?


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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