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How Size and Shape Affects Magnets and Coils

If you place a permanent magnet on a sore elbow, you probably would like to know the magnetic field strength inside your arm where it hurts. Magnetic specs are a good start but additional information on size and shape are needed. From the section on Magnets, you may recall that 2 characteristics are often advertised: Residual Magnetism (Br) and Maximum Energy Product (BHmax). Typical values for common magnetic materials are listed in the table below.


Type

Composition

Residual Magnetism, Br

Max Energy Product, BHmax

Rubber Ferrite in a flexible base

1.5 - 2.5   KGauss

0.5 - 1.5   MGaussOe

Ceramic Ferrite ceramic blends

2 - 4

1 - 4

Alnico Aluminum, nickle, cobalt

8 - 14

5 - 8

Samarian Samarium, cobalt

8 - 10

18 - 27

Neodymium Neodymium, iron, boron

12 - 14

28 - 48


For a given size and shape, the higher these numbers, the stronger the magnet. In medical use, most magnets are round disks. For such magnets without any additional iron shunts, you can calculate the field strength from the graph below knowing just the Residual Magnetism and the dimensions. To allow a single graph to suffice, all dimensions are given in terms of the diameter of the magnet. Let's take an example: your elbow hurts about 1 inch inside your arm and you buy a neodymium magnet 1 inch in diameter and 1/4 inch thick. Then the diameter (D) is 1 inch, the length (L) is 0.25 D and the distance from the pole face (X) is 100% D. Looking at the purple line on the graph, it is a little bit hard to read at 100% D but the magnetic field would be about 2% of the Residual value. Taking the Residual Magnetism from the table as 12,000 gauss (12 KG), the field strength inside your elbow would be about 240 gauss. That's not too bad but it's quite a disappointment after looking at the magnet specs. And this is an expensive magnet. Most magnets are smaller, thinner and ceramic. Performing the same calculation on a 1/2 inch ceramic magnet with a thickness of 1/8 inch, you come up with only 10 gauss inside your elbow.

Field Strength of Permanent Magnet Buttons ( based on diameter )

Permanent Magnet Graph of Gauss and Relative Size

What about electromagnets? Let's consider coils with larger diameters to improve their reach. This can be easily achieved with electromagnets since we are just talking about wire. The disadvantage comes from practical limits that tend to reduce the field strength as the coil diameters grow. A 3 inch diameter coil of 70 turns seems to be a good compromise when it is pulsed with a current of 20 amps rising in 1 millisecond. The graph below plots the strength of the magnetic field on the axis of this coil as a function of the distance from the center of the coil. The axis is the shaft that the coil would spin on if it were a wheel. At a given axis distance, the field is fairly constant up to the inside diameter of coil. The numbers from this graph can be scaled for greater currents. For instance, if the current were doubled, the fields would also double.

Field Strength of an Electromagnet in Gauss

The distance from the coil center is conveniently plotted in cm, where 2 cm is about 1 inch. The coil has an effective reach of up to 6 cm where the field drops to one quarter strength. In the case of your arm, a sore elbow could be placed inside the coil. This puts the coil center right over the target joint. If your shoulder muscles were sore, the coil would have to lay flat and the target would be about 2 cm into your body. In both cases, the peak magnetic field would be 100 gauss or higher. If the coil were pulsed with a risetime of 1 millisecond, the changing magnetic field would produce small electric currents in the fluids of your body. Around the edge of a circle a little bit smaller than the coil, the circulating current would be driven by an electric field of about 0.15 volts/meter. That's not a whole lot but it may be enough to influence body chemistry on the time scale of milliseconds. And such electric fields are a hundred times stronger than anything a permanent magnet could create even if you were rolling around on a matress with dozens of magnetic buttons. This is not meant to be a criticism of permanent magnets. It is meant to contrast the differences between permanent and pulsed magnets. Permanent magnets are inexpensive, convenient and many people swear they offer relief. But, it would appear that permanent and pulsed magnets must affect the body via different mechanisms.

Combined Electromagnet Coils and Resultant Field VectorsTwo magnets may be combined to make stronger or altered fields. Consider the drawings of electromagnet coils shown to the left. If coils are placed parallel to one another like stacked pancakes, the fields in between will add or subtract depending on the orientation of the poles. If the north poles are both facing in the same direction, the fields will add. If a north pole faces a south pole, the fields will subtract. But since the fields drop off with distance, the only point where they match is halfway between the coils. At this midpoint, the additive field will be double or the subtractive field will be zero. The additive case is a good way to double the strength of a field inside a large body part like a knee. Just place the coils on either side of the knee with the north poles both facing to the right. The subtractive case creates what is commonly called a Hemholtz field. Moving between the coils, the field goes from strongly north on one side, to zero in the middle, then strongly south on the other side. If the coils are spaced just right, the gradient is nearly linear like the slope of a ramp. Such coils on a much larger scale are used in MRI (magnetic resonance imaging) machines in hospitals to tag signals with the information on position necessary to create a picture.

If one coil is rotated so they form part of a square, the fields will combine at an angle. The field in the middle is at a symmetric angle of 45 degrees to the original fields and larger by 1.4 times (which is the square root of 2). If combined coils are separated by an inch or more, they are only loosely coupled and can be treated as nearly independent. However, for coils right on top of one another, the coupling is tighter and they may influence each other during the rise and fall times of the field, acting much like additional turns on the same coil. This has the potential of slowing things down and reducing any rate effect circulating currents in the target tissue. Consequently, for most pulsed medical applications, multiple coils would be separated.

What about putting an iron core inside the coils? Intuitively, it would seem like this would be a great way to multiply the field strength. Unfortunately, it is counter productive in medical circumstances. The magnetic field is amplified within the core but falls off very rapidly outside much as the permanent magnets plotted above. Since we need to project the field inside body parts, a core doesn't help much. It also gets in the way of elbows and fingers that could otherwise fit inside the coil. Finally, a core drastically increases the magnetic inertia of the coil slowing the risetime to a crawl. This greatly reduces the rate effect circulating currents which may play a significant beneficial role. Sorry, iron cores are in the trash can.

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