(Magnetism) Magnetic pull force on a steel ball

[amoghthebad]

Homework Statement

Relationship between final velocity of a steel ball and magnetic field strength in a gauss cannon.

Relevant Equations

F = (μ * B^2 * V^2) / (2 * μ0 * d^2)
F = ma
v^2=u^2+2as

I have to write a physics research project, and the topic I chose is magnetism. More specifically, the relationship between the final velocity of a steel ball, and the magnetic field strength in a gauss cannon. However, for the part where I use mathematical formulas to predict the results, things have gone extremely downhill. The problem is that I have already conducted the experiment, and the values that I am getting mathematically are definetly not lining up with the actual experimental values.

The plan was to calculate the magnetic pull force using the first formula: F = (μ * B^2 * V^2) / (2 * μ0 * d^2)
This formula was taken directly from this site, as it seemed to be exactly what I needed: https://www.physicsforums.com/threads/calculate-magnetic-pull-force-for-steel-ball.743775/

μ, magnetic permeability of ferratic steel (the material of the ball) = 1.76*10^-3
V= Volume of the ball(cubic metre, m^3) = 4*10^-6
μ0= Magnetic Permeability of free space (Henries per Metre, H/m) =1.2566*10^-6
d= Distance between the ball and the magnet (metre, m) = 0.075

The values for B (magnetic field strength) were calculated using this calculator: https://www.kjmagnetics.com/fieldcalculator.asp
Orignally, the distance from the magnet was set to 75mm (75 using metric units on the y value in the calculator), however when that didnt work, I set it to the lowest distance I could for each thickness of the magnet, but that also didnt work.

Here is the table of thickness of the magnets with the field strength 75mm away from them

Then, the next step was to substitute the pull force value into the formula F=ma, to calculate the acceleration.
m=mass of the ball (Kilograms, kg) = 0.031

The image above is a picture of the final formula, and how I got it. At this point, I knew something was very wrong, as I realized the value of acceleration would be too small to make sense, and that the first formula (F = (μ * B^2 * V^2) / (2 * μ0 * d^2)) was no good.

The final step was to substitute the acceleration value into the formula v^2=u^2+2as to calculate v, and input v into the equation again in place of u for the second stage of the cannon, and input the result into the formula one last time for the three stages of the gaussian cannon.

However, the values that I am getting are very wrong.

As obvious, the values of acceleration are way too small. As a result, the final velocity obtained is literally no different from the inital velocity.

I am pretty sure that the first formula is wrong, as the values for pull force that i am getting from it are way too small. For example, using the formula, the magnetic pull force value that I get for a magnet 5.001mm thick and 19mm wide (grade N35) is 5.01*10^-4. However using the magnetic pull force calculator from the same website, the pull force value that I get is 59.11 newtons.

So my request is: Can anyone explain to me why the formula isnt working, and what change I could make to the formula to make it work? Or is there another formula that I could use? I could simply just use the calculator from the site instead of a formula to get the pull force values, but I am afraid that the marker will think that it's too simple, and give me a low marking.

 

[amoghthebad]

Another example: The value of the pull force that I gained from the calculator at distance = 7.5mm is
3.15823735 newton

However at the same distance, the value I gained from the formula is 4.91*10^-6 newton. I have no idea why.

For extra information, the diameter of the magnet is 19mm, and the thickness is 5.001mm.

 

[Steve4Physics]

Hi @amoghthebad. Welcome to PF. You are asking a long and messy question which is probably why you've not yet had any replies. Also your experimental setup is not described, so we have to guess.

Here are a few thoughts which might help.

Calculating the acceleration is very difficult for various reasons including:
- the magnetic field changes rapidly (and non-linearly) over short distances;
- therefore so do the force and the acceleration.

So, for example, you can't directly use 'suvat' equations such as $v^2 = u^2 + 2as$ because acceleration is not constant. It is not cear how you have allowed for the changing force (and hence the changing acceleration) in your calculations.

The plan was to calculate the magnetic pull force using the first formula: F = (μ * B^2 * V^2) / (2 * μ0 * d^2)
This formula was taken directly from this site, as it seemed to be exactly what I needed: https://www.physicsforums.com/threads/calculate-magnetic-pull-force-for-steel-ball.743775/

I can't see the formula in the link. If it's correct, it may be inaccurate here because:
- B is not constant inside the ball;
- distance d isn't defined clearly (need to know the 2 exact end-points).

μ, magnetic permeability of ferratic steel (the material of the ball) = 1.76*10^-3

No units are given. Does your equation use the absolute permeability (1.76x10^-3 H/m) or the relative permeability (about 1400)? If your equation uses relative permeability but you have mistakenly used the absolute permeability, then your value of force should be around one million times bigger!

Alternative approach:

If you can, do some additional measurements and produce a force-distance graph for the steel ball and magnets. The area under the graph represents work done and you should be able to estimate the final speed using an energy calculation. This should give you an upper limit (assuming no energy losses) for the final speed.

 

[amoghthebad]

Hi @amoghthebad. Welcome to PF. You are asking a long and messy question which is probably why you've not yet had any replies. Also your experimental setup is not described, so we have to guess.

Here are a few thoughts which might help.

Calculating the acceleration is very difficult for various reasons including:
- the magnetic field changes rapidly (and non-linearly) over short distances;
- therefore so do the force and the acceleration.

So, for example, you can't directly use 'suvat' equations such as $v^2 = u^2 + 2as$ because acceleration is not constant. It is not cear how you have allowed for the changing force (and hence the changing acceleration) in your calculations.


I can't see the formula in the link. If it's correct, it may be inaccurate here because:
- B is not constant inside the ball;
- distance d isn't defined clearly (need to know the 2 exact end-points).


No units are given. Does your equation use the absolute permeability (1.76x10^-3 H/m) or the relative permeability (about 1400)? If your equation uses relative permeability but you have mistakenly used the absolute permeability, then your value of force should be around one million times bigger!

Alternative approach:

If you can, do some additional measurements and produce a force-distance graph for the steel ball and magnets. The area under the graph represents work done and you should be able to estimate the final speed using an energy calculation. This should give you an upper limit (assuming no energy losses) for the final speed.

Hi Steve, I don't know how quote on this site just yet so I will just use numbered points to reply.
1. I accounted for the rapidly changing force by taking distance as 1/4th of the value between the two magnets (30cm), so 1/4th of that comes out to be about 0.075m.
2. The formula is in one of the replies, I needed to scroll to find it.
3. I don't know. The link just said "Magnetic Permeability of the steel ball", did not mention absolute or relative. I assumed it meant absolute, so i used that. Also have no idea what happened to the reply, it seems to have been deleted, no idea why. I might try again, using the relative permeability this time instead of the absolute one.
4. Thank you so much for this suggestion, I will probably use this idea to find out the kinetic energy gained, and use that to calculate the final velocity. Thank you so much! I really appreciate your time.

 

[berkeman]

Hi Steve, I don't know how quote on this site just yet so I will just use numbered points to reply.

I will send you some hints on this via Private Message (PM).