The vortex electric field of a solenoid

in accord with eq. 1.188, the vortex field inside the solenoid (r a) is  

That is, the electric field inside the solenoid increases linearly with the radius r. 

Considering now horizontal circles located outside the solenoid, it is clear that the flux and its time-derivative do not depend on the radius of the circles, and that at any given time, we have  

The voltage along any of these circles is therefore also independent of the radius and according to eq. 1.188, we can write  

That is, the vortex electric field outside the solenoid decreases inversely proportional to 

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