I am working on a medical device application and would like to use a solenoid with a variable turn density such that the force \$F\$ on a metal piston within the solenoid cylinder is not constant but is proportional to the the overlap length \$l\$ between the piston and cylinder - effectively functioning as a spring with spring constant as a function of the current.
$$ F(l)=k(I)l $$
My naive assumption is that I can make this work by using a wire turn density which is a function of the overlap \$n(l)\$, instead of the usual uniform turn density \$n=N/L\$, where \$N\$ is the total number turns and \$L\$ is the total length of the solenoid. I made some progress using the Ampere derivation of the magnetic field on infinitesimal slices of the cylinder but couldn't figure out how to do the line integral. Any insights would be much appreciated.