Electromagnetics question for physicists/engineers

[jjb23]

I've been looking into electromagnet design and there is a certain principle/set of equations that are bothering me...

(1) B = (kμ₀Ni)/L

(2) ø = BA

Where;
- ø = total magnetic flux (wb)
- B = magnetic flux density (wb/m²)

I am trying to solve for an optimum cross sectional area of core material in a solenoid, however according to these equations flux density (field strength) is not proportional to cross sectional area. So in theory you could keep increasing the diameter of your core material to increase the total flux with all other variables staying constant.

I would have thought that total flux was a conserved quantity with flux density reducing with increased area. This does not seem to be the case, and makes little sense.

As a physical analogy, it seems like having a material of constant density and increasing the volume to increase the mass... breaking every rule in the book!


I may well be missing something somewhere along the way, if you could advise as to a solution to my cross sectional 'core area' problem that'd be great.

 

[Simon Bridge]

When you increase the crossectional area, you are increasing the length of wire in each loop too - giving more moving charges surrounding the area.

[edit] sorry, had to step out for a bit... to continue.
I realize that the circumference increases as L while area increases as L^2 - so including that idea into the argument suggests that flux density should depend (inversely) on the radius of the loops - which is what happens for a single loop.
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html
... getting closer. Extend that line of thinking for a large number of loops ... the flux density will have an extra contribution from the number of loops - adding in an extra length dimension in the numerator.

To really see what's going on, you need to do the derivation.
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

 

[jjb23]

Thanks for your reply.

I've already considered how coil length would increase with increased core diameter, but this only means that the flux density will remain constant at the surface of the core regardless of diameter. I have also done the derivations (several times!) and it still seems odd to me.

Anyone got any suggestions/equations for core area?

 

[Simon Bridge]

Well if it still seems odd after doing the maths, and after building one and measuring, then there is nothing we can do. You will just have to accept that it is odd.

 

[jjb23]

Perhaps some trail and error will yield some relationships. It does seem a bit of a grey area; I may have just found myself a topic for my research paper, cheers.

 

[Simon Bridge]

You already have the relationships - you just find them odd.
But if you have not done any experiments on this before, you should certainly do them.
See if the magnetic field strength in the center of the solenoid (what the formula is for remember) depends on it's crossection area.