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# Physics/Problem calculating magnetic field strength

Question
Hello Steve,

Thanks very much for your previous answers, they were very helpful for me and I really appreciate the time you take.
I am interested in calculating the magnetic field strengths of electromagnets with a couple of different core geometries, and I have been making some calculations that I think have gone wrong somewhere and I was just wondering if you might see fit to kindly point out where/how I have made a mistake.

I am trying to calculate the field strengths of a straight cylindrical core electromagnet and also a U shaped electromagnet. I have assumed that the U shaped electromagnet is going to produce a stronger field, given the poles are closer and the magnetic field has less distance to travel through free space than the cylindrical core electromagnet.
However, when I do the calculations, I get a lower value (in Tesla) for the U shaped electromagnet than the straight cylindrical core electromagnet. Either my initial assumption the U shaped electromagnet would be stronger was incorrect, or my calculations are wrong (given my lack of scientific knowledge, either are pretty likely!)

For both magnets, the number of turns, the length of the core, the current, and the core material permeability are the same values. (Both are very small magnets)

Number of turns = 6
Core length = 0.0152 meters
Current = 0.0001 Amps
μ Core Permeability = 50
μ0 Free space Permeability = 0.0000012566

For the straight cylindrical core electromagnet, I used the formula I got from here http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

permeability*turn density*current = B

so I did the following calculation
permeability (50 * 0.0000012566) * turn density (6 / 0.0152) * current 0.0001 = 2.48013158 × 10-6 Tesla

For the U shaped electromagnet, I used the formula I got from here http://en.wikipedia.org/wiki/Electromagnet
Under the heading "Magnetic field created by a current" it says "For an electromagnet with a single magnetic circuit, of which length Lcore of the magnetic field path is in the core material and length Lgap is in air gaps, Ampere's Law reduces to:

turns*current = B((Lcore/μ)+(Lgap/μ0))

Which I changed to
(turns:6 * current:0.0001) / ((Lcore:0.0152 / μ:50) + (Lgap:0.0008 / μ0:0.0000012566)) = B = 9.4244955 × 10-7 Tesla

9.4244955 × 10-7 Tesla for the U shaped electromagnet vs 2.48013158 × 10-6 for the straight cylindrical core electromagnet seems incorrect to me. I feel like, if anything, it should be closer to the other way around. I have checked each equation using an online calculator, with Excel, and through simply typing each equation into the google search window
eg
(50 * 0.0000012566) * (6 / 0.0152) * 0.0001 =
and
6*0.0001/((0.0152/50)+(0.0008/0.0000012566)) =

BUT I can't seem to find where I have gone wrong. Do these results seem strange to you? Am I using the wrong formulas perhaps?
Any suggestions would be most welcome.

Thanks again for all your patience!

OK, first and foremost, keep in mind that all these formulae are for on-axis, center of the magnet field, and they assume impossible ideal conditions.  Including the permeability.  And that you have to integrate teh fields.  You used two different sources for equations.

That given, let me look at your parameters.

1)  Your number of turns are too small, as are your core lengths.  Those don't apply to solenoids (length = long vs diameter of turns, and a large number of turns per unit length).
2)  You're only off by a factor of 2.5, which isn't wildly off, given the assumptions made.
3)  You made not one, but several math errors.  Basically, you can factor out turns and current, since they appear in the numerator in both equations and create an equation for B/IN=1/((Lcore/μ + Lgap/μ0)) for the u-shaped magnet and B/IN=50μ0/L for the solenoid, where L is the length, I is the current, and N is the number of turns.  In this case B/IN is still dominated by the gap for the U-shaped magnet, where it doesn't even factor in to the ideal (which this is NOT) solenoid.

Basically, your geometry is wrong for the approximations of these equations.  Yes, approximations.  You MUST have a long, thin solenoid for these equations to work, and a high number of turns per unit length.  Neither equation accounts for the current that is effectively travelling along the coil.  Six turns is a ridiculously number of turns for a magnet, these numbers typically start in the tens of thousands for magnets that obey these equations.  Your core length is only 1.5 cm, which still means that your core gap is very large relative to this length.

Good luck, you stumbled onto a very complex problem.

Physics

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#### Dr. Stephen O. Nelson

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