# Difficult problem about eddy currents and terminal velocity

1. Jul 23, 2010

### Ghengis

Hey guy,

Ive been recently investigating the phenomena of eddy currents, particularly that of a magnet falling down a copper tube.

To the people who havent seen it before, basically the magnet reaches terminal velocity in the copper tube extremely quickly, and takes a long time to leave the copper tube.

Anyway, i was wondering how could you show that the time taken for the magnet to reach terminal velocity is negligible.

Thanks so much guys :)

p.s. i have already calculated the formula for estimating the final velocity of the copper magnet, but im stumped on figuring out how to prove that it would reach the terminal velocity in a very small period of time.

Last edited: Jul 23, 2010
2. Jul 24, 2010

### vela

Staff Emeritus
Come up with the equation of motion for the magnet and solve it. Do you have an expression for the magnetic force as a function of its velocity?

3. Jul 24, 2010

### Ghengis

i know that the downward force on the magnet is mg
i know that the upward force on the magnet due to the eddy currents is (B^2 times L^2 times v) divided by R where B is the tesla, L is the length of the eddy current, v is the velocity and R is the resistance.

From here, how do i show that how much time is taken for the upward force to equal the downward force?

Thanks

4. Jul 24, 2010

### vela

Staff Emeritus
Use Newton's second law and solve it for v(t).

5. Jul 24, 2010

### Ghengis

Oh, i get it!

so just to double check

F=mv/t
and F=mg (in this case)

mg = mv/t
g=v/t
t=v/g

Is that right?

6. Jan 1, 2011

### ScottTheSculp

Ghengis!
I just joined up and was looking for clues on how to calculate the same thing.
I'm a sculptor working on an eddy current project and intend to engrave the proper equation onto the sculpture.
Could I possibly get a copy of your equation?

I think I sidetracked too much on trying to integrate the flux through the cylinder walls.

ScottTheSculptor
Nerd Art

7. Jan 1, 2011

### vela

Staff Emeritus
Oops, looks like I lost track of this thread. Sorry, Ghengis.

For the record, Ghengis's equations are not correct. For one thing, the force due to the eddy current doesn't appear in them. Also, the acceleration isn't equal to velocity divided by time; it's the derivative of velocity with respect to time.

8. Jan 1, 2011

### ScottTheSculp

thanks vela,
I haven't found *any* complete description of the slowing of a magnet falling in a non-magnetic conductive tube.
Parts are described - always simplifying to avoid the calculus of integrating the induced field over time and distance.
I'm currently wrestling with a paper on the dampening of an oscillating magnet in a pipe - best info so far.
Though difficult . . .

ScottTheSculptor

9. Jan 1, 2011

### ScottTheSculp

http://en.wikipedia.org/wiki/Magnetic_damping

the "differential" equation is wrong.

no acceleration of the magnet, so Mg=KV?

or

Whole page is pretty much nonsense?

As an artist I'm not quite confident in math and physics to call them on it.

ScottTheSculptor

10. Jan 1, 2011

### ScottTheSculp

Woohoo!

http://www.if.ufrgs.br/~levin/Pdfs.dir/AJP000815.pdf

All the math is there, good description on the derivation

in case the above link goes invalid;

American Journal of Physics
Volume 74, No. 9, September 2006
"Electromagnetic braking: A simple quantitative model"

11. Jan 2, 2011

### Ghengis

Vela, dont worry, i knew that it was the derivative. I wrote it properly in my paper.

Scott, the equation they give is extremely complex and although i used that paper as a launching pad, i didnt know how to use the scaling functions, so i had to derive my own. if you are still interested, the formula i obtained was:

Terminal Velocity = (mass of magnet x gravity x resistivity of metal pipe) DIVIDED BY (2 x Magnetic field strength squared x pi x radius of copper pipe squared x thickness of copper pipe)

This formula is relatively accurate. The data agreed to 10 percent of this value!

12. Jan 2, 2011

### ScottTheSculp

Thanks!

I like the abstraction of the magnet into two sides that oppose the fall.

For engraving purposes the math can be far more complex than I can actually work :-)

Did you work out the terminal velocity time?
I don't see how to show time in the speed that the induced current induces the opposing magnetic field so figure that the whole thing works as fast as the magnet mass is accelerated by gravity to terminal velocity?

I'd like to see this modelled in super slow motion CFD with thick walled pipe.
The actual currents in the pipe are hard to imagine.
inverse square action from inside to outside radius with the fields of the inner currents messing with the path of the outer currents and the fields from the outer currents slighly modifying the inner and fields rotating around all those currents. whew.

13. Jan 3, 2011

### Ghengis

haha! what if someone asks you how you proved the formula engraved on the sculpture? :D

Ya, i did work out the terminal velocity time.
This is because i made the assumption that the magnet almost instantaneously reaches terminal velocity when it enters the magnet. I proved this mathematically. If the magnet is strong enough, the time taken for the magnet to reach terminal velocity is something like a millisecond. Thus its pretty easy to calculate.

I dont think anyone has actually been able to properly visualize those currents. Because the magnetic field strength is highly unlikely to be uniform, the eddy currents are just plain out weird. That is why no one has been able to make a very accurate formula.

14. Jan 3, 2011

### ScottTheSculp

This whole project started by accident.
I tossed one of my clamp magnets in a copper casting and it didn't fall right.
Took nearly 15 seconds of dumbfoundness before I remembered the physics.
Thinking to turn it into a sculpture I thought about the math and immediately realized how complex the problem is.
Assuming someone else had already figured it out I headed for the Fog ( 'net, cloud, whatever). As usual, I expected more knowledge than is there.
But I have found reading the explanation for calculations that only yield approximations to be a way into these complex systems.

I can visualize the 3D physics much better than I can do the math :-)
I now plan to engrave the proper Laws and Calculus without reducing them into a solvable equation.
That's the beauty of a really complex problem and art - don't solve it, just show it.