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Hi! How can I calculate the magnetic force on an object, such as bismuth? Is that force on the entire object or on just one particle?
Question Date: 2008-06-27
Answer 1:

The mathematics is not simple, so if you have bismuth and a magnet, I recommend directly measuring it instead. Put a chunk of bismuth on the front of a cart with low rolling resistance (Legos work nicely) on a flat plane (cookie sheet?) and push it uphill with a magnet. Raise the angle of the inclined plane until the magnet is just unable to push the cart uphill. Measure the angle you had to raise the inclined plane. This gives you the pushing force.

Start with Newton's law:
where m is the total mass of the bismuth+cart.

Since gravity is our accelerating force:
where g=9.81 m/s2 on the surface of Earth.

But this formula needs one more major correction. Gravity is pulling downward (vertical), but the force between the bismuth and the magnet is only horizontal, and only partly vertical. How much gravitational force is along the incline? From basic trigonometry (which you may have seen already), the *fraction* of force is the sine of the angle from horizontal:
F (along the direction plane) = F (vertical) * sin (angle of plane from horizontal)

For example, the sine of 5 degrees is .0872. So a 100g cart on a slope of 5 degrees experiences
F = 0.1kg * sin (5 degrees) * 9.81 m/s2 = 0.1 * 0.0872 * 9.81 (kg*m/s2) = 0.085 Newton of force down the hill. (1 Newton = 1 kg*m/s2)

So if your bismuth-magnet combination is able to just barely repel a 100g cart at an angle of 5 degrees, you know the repulsive force is 0.085 Newton

You can thus experiment with different combinations of magnets, and the size & shape of bismuth, to see which gives the strongest repulsion. Once you find the configuration which provides the maximum amount of force, you can calculate how much stronger your magnets would need to be to levitate the bismuth. (Levitation would be when the angle=90 degrees, so sin (90 degrees) =1.)

(A caution when calculating sines: calculators and spreadsheets like AppleWorks or Excel sometimes use radians instead of degrees. So try calculating sin (90) and see if you get 1.00. If you get 0.893 instead, then the calculator or spreadsheet is using radians. Fortunately, the conversion is simple: radians = degrees * pi/180, or approximately degrees/57.)

Galileo did a lot of work measuring forces and accelerations with an inclined plane. You'll be following in the footsteps of a master!

Answer 2:

I do not understand solid-state magnetism, so I can't help you if you intend to repel your bismuth sample using a normal magnet. However, I can help you if you are using an electromagnet (which is physically a great deal simpler).

If you have a long, straight wire that is carrying current, then the magnetic field that it will generate will be B = 2KI/d, where K is a proportionality constant (exactly 10-7 in SI units), I is current (in amperes), and d is the distance to the wire (in meters). The direction of the magnetic field at any point will always be perpendicular to the wire and given by a right-hand rule.

You can then apply the formulas that I was able to find for you last time that will give you the strength of the diamagnetic reaction of Bismuth to the current in the wire, since you will now know the strength of the magnetic field.

If you're still planning on using a ferromagnet, then I can't help - this is beyond my understanding of physics.

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