Constant Field Throughout Entire Solenoid Axis

[RDBaker]

Along the center of a very long solenoid tube the field strength is constant. At the ends of the tube, however, the field decays and drops off.

Is there a simple way (mathematical solution or simple trick) to make the field perfectly constant (or to a fraction of a percent tolerance) throughout the whole volume of the solenoid?

I have two ideas to solve this problem:

[1] Add extra layers of wires at the ends.

This should work but adds more wire, increasing the power requirement. I also haven't modeled this one yet.

[2] Vary the radius of the tube from the central axis.

When I model a solenoid with an convex parabolic contour (a guess) I get some pretty good results for about 83 percent of the tube.
I'm working on a more rigorous mathematical way to derive the optimal solenoid shape. Any thoughts would be much appreciated.

 

[berkeman]

Along the center of a very long solenoid tube the field strength is constant. At the ends of the tube, however, the field decays and drops off.

Is there a simple way (mathematical solution or simple trick) to make the field perfectly constant (or to a fraction of a percent tolerance) throughout the whole volume of the solenoid?

I have two ideas to solve this problem:

[1] Add extra layers of wires at the ends.

This should work but adds more wire, increasing the power requirement. I also haven't modeled this one yet.

[2] Vary the radius of the tube from the central axis.

When I model a solenoid with an convex parabolic contour (a guess) I get some pretty good results for about 83 percent of the tube.
I'm working on a more rigorous mathematical way to derive the optimal solenoid shape. Any thoughts would be much appreciated.

Welcome to the PF.

What is the source of the magnetic field in this problem?

Can you give us some more context to the setup? What are you wanting to achieve?

 

[RDBaker]

Thanks for the welcome!

A solenoid electromagnet is usually a hollow cylinder wrapped with wire. Someone who wants to make a magnetic field in the middle of that cylinder needs to run current through the wire around it. The magnet strength isn't uniform inside the tube though, because near the very edge the magnet strength (field) begins to weaken. I would like to make a shape that I can wrap wire around so as to make the magnet strength uniform inside the shape. I can add more wires to the end or I can change the shape.

The way I have looked at the problem so far is that I have taken the field generated by one loop of wire (from the Biot-Savart Law), and made a computer program to add many fields together to describe the field strength inside an arrangement of wire. I hope to find a shape (Wire Configuration) which forms a perfectly uniform field inside the volume of the shape.

 

[Bob S]

This equation for the on-axis field is exact both inside and outside the solenoid:

http://www.netdenizen.com/emagnettest/solenoids/?solenoid

You can add more layers at the ends.

Bob S