I have a Neodymium magnet, for example, that has a magnetic flux density of $1.25 \;\text{T}$ and is 1 inch by 1 inch by 0.44 inches. I have a steel ball with a diameter of 0.5 inches placed 3 inches away from the magnet. I want to know how much force is acting on the steel ball. How can I do this? Is there some equation or method to calculate this?
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$\begingroup$ Why don't you perform the experiment. Put the ball above a low-friction surface and use the magnet to attract the ball. The magnet will magnetize the ball and it will attract it. Use Newton's law of motion to find the missing force(you can ignore air drag) which will be the magnetic force. $\endgroup$– Jun Seo-HeCommented Jul 19, 2021 at 21:08
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The formula you can use is $$F=\frac{B^2A}{\mu_0}$$ where the permeability of a vacuum $$\mu_0=4\pi \times 10^{-7} Hm^{-1}\approx 1.26\times 10^{-6} N\ \text{amp}^{-2}$$
This is an approximation and the problem with this is the area $A$ is usually that which is either close to, or in contact with the magnet (this is the same as the force needed to separate them), and here the area is spherical.
Since the magnet is fairly small, and the distance of separation is also small, you can use the projected area of the magnet facing the ball (i.e., half its surface area) and this will give you a close value to the experimentally determined result.
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$\begingroup$ So the area is the surface area of the magnet or the ferromagnetic material? Also, why does the distance between the magnet and the ferromagnetic particle not in this equation? $\endgroup$ Commented Jul 22, 2021 at 0:23
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$\begingroup$ Yes. This equation assumes that magnet/ferromagnet is in contact with the magnet, or at least very close to. This is why surface area is proportional to the force. Cheers. $\endgroup$– joseph hCommented Jul 22, 2021 at 0:26
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$\begingroup$ But the surface area is of the ferromagnetic thing or the magnet itself? Thanks! $\endgroup$ Commented Jul 22, 2021 at 0:35
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$\begingroup$ Ideally, the area A is that which coincides with magnet and substance. Remember that I gave you this equation as an approximate method to calculate the force, and it will not be straightforward to do this since the ferromagnetic substance is spherical. If the magnet’s area is larger than the sphere, take the projected surface area of the sphere facing the magnet. Technically a very small area of it (being a sphere) is actually in contact.If the sphere is larger than the magnet, you would use the flat surface area of the magnet facing it.Let me know how close to experiment you get for F.Cheers. $\endgroup$– joseph hCommented Jul 22, 2021 at 1:19
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$\begingroup$ I just remembered that you actually give out values for area, diameter etc after reading your question again just now. The comment above still applies, though you do have numerical values to use. Cheers. $\endgroup$– joseph hCommented Jul 22, 2021 at 1:24