# Trouble understanding fractional flux linkage

My question is quite similar to one in the link: Flux linkage inside of a conductor. But I have trouble understanding any of the answers provided there. Specifically my question is the following: I need to calculate the internal inductance due to the flux linking inside the wire. If I consider Faraday's equation I can write $$L\frac{dI}{dt} = \int {\bf E}\cdot{\bf dl} = \int {\frac{\partial {\bf B}}{\partial t}\cdot{\bf ds}} .$$ To evaluate the potential difference that develops across unit length along the conductor let me consider a loop that is starting from the center of the conductor and returns through the inner surface. If I use $${\bf B} = \frac{\mu x I}{2 \pi r^2},$$ I get $$L\frac{dI}{dt} = \int_0^r \frac{\mu x}{2 \pi r^2}\frac{d I}{dt} dx .$$ But this gives the inductance per unit length as $$L= \frac{\mu}{4\pi} ,$$ which is double the actual value. I know that this is because "fractional linkage" is not considered but I do not understand a logically consistent way to introduce this concept into Faraday's equation. What am I missing ?