Air core transformer

Phrak

I wind a solenoid 1/2 inch diameter and 10 inches long on a plastic rod. A secondary winding consists of a single loop of wire at the center of the solenoid that's a little over 1/2" in diameter. If I put 1 mVpp per turn on the soleniod I'd expect about 1 mVpp on the secondary, as if it were an auto-transformer. About what voltage do you get if the diameter of the secondary is changed to 1 inch diameter?