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$$ \overrightarrow{\bigtriangledown } \times \overrightarrow{B} = \mu_{0} \left [ \widehat{J} + \varepsilon _{0} \frac{\partial \overrightarrow{E} }{\partial t}\right ] $$

So calculating the magnetic field B, around an infinite line of current, where the current flows in the positive z direction, using cylindrical coordinates, I get:

$$ \overrightarrow{B} = \frac{\mu _{0} I}{2\pi \rho }\widehat{\phi} $$

Then to find the change of the electric field in the vicinity, take the curl of B, and we get:

$$ \overrightarrow{\bigtriangledown } \times \overrightarrow{B} =

\left [ \frac{1}{\rho } \frac{\partial Bz }{\partial \phi } - \frac{\partial B\phi }{\partial z} \right ]\widehat{\rho } +

\left [ \frac{\partial B\rho }{\partial z} - \frac{\partial Bz }{\partial \rho } \right ]\widehat{\phi } +

\frac{1}{\rho }

\left [ \frac{\partial (\rho B\phi) }{\partial \rho } - \frac{\partial B\rho }{\partial \phi } \right ]\widehat{z } = 0$$ which seems like it makes sense because the current and magnetic field are not changing. But, if i use a time varying current: where $$ I= I_{0}cos(\omega t) \Rightarrow \overrightarrow{B} = \frac{\mu _{0} I_{0}cos(\omega t)}{2\pi \rho }\widehat{\phi} $$ but i still get:

$$ \overrightarrow{\bigtriangledown } \times \overrightarrow{B} =

\left [ \frac{1}{\rho } \frac{\partial Bz }{\partial \phi } - \frac{\partial B\phi }{\partial z} \right ]\widehat{\rho } +

\left [ \frac{\partial B\rho }{\partial z} - \frac{\partial Bz }{\partial \rho } \right ]\widehat{\phi } +

\frac{1}{\rho }

\left [ \frac{\partial (\rho B\phi) }{\partial \rho } - \frac{\partial B\rho }{\partial \phi } \right ]\widehat{z } = 0$$

So even though I have a time varying current, which seems to produce a time varying magnetic field around the infinite wire, I don't seem to be getting a time varying electric field as I would expect. I can understand why mathematically, seeing that I

_{0}cos(ωt) is not a function of ρ or z, so that in the curl, the partial derivatives with respect to ρ and z are still 0.

So I'm not sure where I went wrong. I can see that if I included a phase term in the current, where I

_{0}cos(ωt-βz) the curl of B would not equal 0, but I thought that just a time varying current would cause a changing magnetic field which in turn would cause a changing electric field. Can someone point my in the right direction about where I'm going wrong? Thanks