B field of 1/2 infinite solenoid, equivalent current confguration

[Spinnor]

I have wondered if there is a symmetric current configuration that gives the magnetic field of a half-infinite solenoid. With some thought I think I came up with such a configuration of current loops that produces the same magnetic field as a half-infinite solenoid

Suppose we have a large but countable number of half-infinite solenoids whose ends all begin at some origin and that point in random directions. Consider each solenoid to be the sum of a large but countable number of very small current loops one stacked on top of the next.

If one half-infinite solenoid gives a radial magnetic field then so does the superposition of any number of half-infinite solenoids with random orientations as long as their ends all end at the origin. So we have a large but countable number of solenoids each composed of a large but countable number of very small current loops. This leads to the following picture.

Consider the magnetic field given by a large but countable number of very small current loops whose orientation all point away or towards the origin and whose density in space varies as 1/r^2 where r is the radial distance from the origin. There is an equivalent configuration, I think, where the loops are randomly distributed in space (again, all orientated towards or away from the origin) but the current in the loop goes as 1/r^2.

Does that seem right? Thanks.

 

[Charles Link]

Your post is interesting, but I don't know that anything with a different current configuration will duplicate the magnetic field from a semi-infinite solenoid. $\\$ The semi-infinite solenoid has the same current configuration as a semi-infinite uniformly magnetized cylinder and has a single pole (surface) on the end face. By comparing the results from the pole model vs. surface current for a semi-infinite cylinder of uniform magnetization, and showing the results for $B$ to be equal outside the cylinder, I was able to prove, with a couple additional steps, the magnetic pole model equation $B=\mu_o ( H+M)$. I did this calculation several years ago. Most E&M texts elude to the analogous electrostatic equation $D=\epsilon_o E+P$, but this is really very much a handwaving argument, and is very incomplete. I was very glad to finally prove this equation, and I was actually surprised that the $B$ computed from the pole model is in 100% agreement with what the magnetic surface currents and Biot-Savart give. $\\$ I realize that my reply may be taking this off on a tangent. The Moderators should feel free to delete my post if they think it is not a suitable or relevant reply.

 

[Spinnor]

I don't know that anything with a different current configuration will duplicate the magnetic field from a semi-infinite solenoid.

Two such solenoids whose ends coincide and can point in any direction (and with the same current orientation, left handed or right handed). A radial field plus a radial field gives you a radial field. Now keep adding solenoids. Yes?

 

[Charles Link]

Two such solenoids whose ends coincide and can point in any direction (and with the same current orientation, left handed or right handed). A radial field plus a radial field gives you a radial field. Now keep adding solenoids. Yes?

This looks like it is basically headed in the direction of a magnetic monopole. See this previous thread that discusses the structure that a magnetic monopole would have: https://www.physicsforums.com/threa...o-vector-potential.950053/page-2#post-6017074

 

[Spinnor]

So I am wrong. If we had a countable number of solenoids emanating from the origin then there would be just as many field lines going in as going out, no monopole field. Back to the drawing board.