magnetic field outside a solenoid

so a solenoid is hooked up to an ac signal. The solenoid core (say ferrite) is much longer than the solenoid coils. How does the magnetic field outside the coils, along the axis of the core look like. Can the attenuation of the magnetic field be explained analytically?
From intuition the attenuation would be exponential but why? obviously inside the coils the field suffers no attenuation.

anyone can explain the physics here?

How it looks like:

Magnetic Fields and Magnetic Forces
**broken link removed**

thanks, but i'm more interested to know if the field outside of the solenoid coils can be explained analytically. The closest example i can think of would be a loop of current and using biot savart law to find the attenuation (see image). This only applies to the magnetostatic case and a single loop. Im looking to describe this for a time varying current and a solenoid.

See this materials maybe can help you:

The magnetic field of a solenoid and a toroidal coil
https://physics.aalto.fi/pub/kurssit/Tfy-3.15xx/Tp_ohjeet/33en.pdf

An Analytical Calculation of the Magnetic Field Using the Biot Savart Law
https://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172001000300005

well biot savarts is only for the magnetostatic case. I'm looking to see how it would work for a time varying current. Maybe ham radio ppl and antenna experts would be able to explain this, if there are any on this forum.

you might want to do a search for "time varying current in solenoid"

the approach in this case, is to calculate vector potentials and proceed directly with maxwell's equations, instead of using biot-savart law (which is a limiting case). [1] should provide some good information

[1] https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.210.3172
[2] https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.8156

well biot savarts is only for the magnetostatic case. I'm looking to see how it would work for a time varying current

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As long as the dimensions are small compared to wavelength, this is a simple AC magnetic problem. If core losses are neglibible, the field is identical to the magnetostatic case. But as soon as cores come into play, biot savart doesn't work any more.

The solenoid core (say ferrite) is much longer than the solenoid coils. How does the magnetic field outside the coils, along the axis of the core look like. Can the attenuation of the magnetic field be explained analytically?

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I don't think that field intensity will follow a simple exponential function. For a rotationally symmetric case, it's a quasi 2-D solution, but still involves complex mathematic. You might find it derived in a theoretical electrical engineering text book, these days most people are using finite element solvers to calculate it, me too.

Intuitively, the core field outside the coil will decrease towards the shutting faces because part of the "field lines" is taking a more direct way around the coil. The field form is similar to a cylindrical permanent magnet extended by straight pole-pieces.

thanks for your replies. FvM, when you say that

If core losses are neglibible, the field is identical to the magnetostatic case

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what do you mean? I'm trying to find the attenuation of the field outside the coils, so the only losses are as a result of the core, or not?

I have already solved this problem with FE analysis, and as you say the field lines look like those for a

cylindrical permanent magnet extended by straight pole-pieces

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and the core is cylindrical for my example. If you say this problem involves complex mathematics, can you explain the starting points or the algorithm required to solve this. This is where I am lost. jeeudr, calculating vector potentials is not my expertise and the papers you provided do explain a similar problem, but not quite the same. If anyone could find an engineering text book with this problem, that would be very helpful.
oh and yes, the dimensions are small compared to wavelength for this problem.

what do you mean?

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That you don't have to worry about "time varying" in a first order.

If you say this problem involves complex mathematics, can you explain the starting points or the algorithm required to solve this.

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Unfortunately I can't.