Calculating Gauss of electromagnet

[JCBII]

It's been many years since I studied this so I would appreciate some confirmation on the formula and its parts.

The formula I was taught to calculate the Gauss of an electromagnet was: B=MNI/2R
with: B = Magnetic flux in Gauss
M= Magnetic permeability of core material
N = Number of turns of wire
I = Current input in Amps
2R = two times radius (or diameter) of core

Question... is this formula correct?

I have noticed that the term Tesla is favored over Gauss. As I recall, 1 Tesla equals 10,000 Gauss. Am I correct?

I have been quoted by a metal manufacturer that the magnetic permeability of Nickle-Iron is 1,000,000 (one million). That same manufacturer said that NdFeB was 900,000. Do these numbers represent μ? Are these figures proper inputs for the above formula?

So let's create a hypothetical electromagnet. We'll use a .25" diameter by 12" long Nickle-iron rod as the core.

Using 24 Ga magnet wire (.0201") we can wind a coil 12" long by 7" in diameter. This would equal 220,000 turns of wire. If we then supply it with 500 mA of current it should look like this:

B = 1,000,000 x 220,000 x .5 / .25

B= 110,000,000,000 / .25

B = 440,000,000,000 or 440 Billion Gauss


Here's my problem:
The Mag Lab outside Tallahassee Florida claims to have built the world's strongest electromagnet which is 45 Teslas (450,000 Gauss) in strength.

So where have I screwed up in this calculation?

 

[marcusl]

Well, I'm afraid that you messed up virtually every principle and aspect of magnet design.

1. 220,000 turns, at a 2.6" geometric mean diameter, corresponds to over 150k feet or 30 miles of 24 gauge wire. Can you could get a single spool or are you going to create joints?

2. The resistance of this coil is 3900 ohms which implies a 1950 V potential across the coil. This is far (really really far) above the insulation breakdown voltage for magnet wire.

3. The power dissipated is approximately 1 kW, which would quickly melt your coil (except for the fact that it would arc over first).

4. Your turns calculation did not include the thickness of insulation, the finite ability to pack wires tightly, or room for cooling tubes and potting compound.

5. The formula for the field inside a long thin solenoid is B=\frac{\mu_{rel}\mu_0NI}{length}. Even this formula, however, does not apply to a very thick solenoid such as you have specified.

6. You messed up the units. The formula above is correct for SI units (mks, and tesla).

7. Special alloys like moly-permalloy have a relative mu of close to 1 million, but are completely useless for this application due to their incredibly low saturation field. You'll need to use iron.

8. Iron saturates at the relatively high value of 10 Oe applied field, which is still quite low. At which point its effective permeability approaches 1. The art in magnet design is balancing iron and windings so that most of the iron remains unsaturated.

In short, the magnet that you propose is unworkable and the field that you calculated is seriously in error.

 

[JCBII]

Thanks for the input Marcus. As I said in my post, this was a hypothetical electromagnet. It's obvious that it's unworkable. Common sense and the claims of the Mag Lab show that. What I was looking for was a correction to the formula so that I could accurately design a high power electromagnet. But it sounds as though there is no such formula. You mentioned a large number of design variables to be taken into consideration. It sounds to me that the whole design process is just too complex for me to tackle. Thank you for putting me in my place.

 

[marcusl]

To get a sense of the scales involved in high field magnets, take a look at this page
http://www.lakeshore.com/products/Electromagnets/Models/Pages/Overview.aspx
These nearly 2 T (20kG) electromagnets weigh 500 - 1500 lbs, depending on the size of the sample volume, and require a large power supply and chiller. The weight comes partly from the heavy copper coils (blue in the pictures) and otherwise from the massive iron cores and yokes needed to convey flux without complete saturation. The 2T figure is not arbitrary--it is the saturation induction of iron. Moving higher requires other approaches.

The ultrahigh field magnet lab magnets that you referred use multiple stages of nested superconducting windings, typically without ferromagnetic cores. The sample volumes are generally very small at the highest fields.