## Magnetic damping

### Magnetic damping

I have seen demonstrations where a magnet dropped through a copper tube, or a metal sheet dropped through a magnetic field, falls slower than normal, due to eddy currents being induced in the tube and sheet respectively. Is there any way to quantitatively describe the motion of the magnet and sheet when dropped? If so, how?

### Re: Magnetic damping

The current density induced would be a complicated function of position, so I think that only a numerical solution would be possible – or at least practical.

An order of magnitude estimate would be less work: make a simple geometric model of the induced current circuit, give it plausible values, look up conductivity of metal (a strong function of impurity content, unfortunately), apply Faraday's law.

Best

Joe

An order of magnitude estimate would be less work: make a simple geometric model of the induced current circuit, give it plausible values, look up conductivity of metal (a strong function of impurity content, unfortunately), apply Faraday's law.

Best

Joe

### Re: Magnetic damping

Wikipedia here ( http://en.wikipedia.org/wiki/Magnetic_damping#Equation ) is giving this differential equation of motion of a magnet dropped vertically through a conductor:

ma = mg - kv, where m = mass of magnet, a = acceleration of magnet, g = acceleration due to gravity, k is a damping coefficient and v = velocity of magnet.

This differential equation's solution is v(t) = (mg/k).[1 - exp(-kt/m)] (if v = 0 when t = 0).

This would mean that the magnet quickly approaches a terminal velocity and then moves at an almost constant speed, asymptotically approaching this terminal velocity.

This appears to match what happens when the magnet is dropped in real life, so does that differential equation have any merit? (It implies that the force opposing the magnet's motion would be proportional to its velocity, would this be a reasonable approximation?)

Also, in which direction do the eddy currents flow in the copper tube? Do they flow around its circumference, or do they flow parallel to the magnet's fall?

ma = mg - kv, where m = mass of magnet, a = acceleration of magnet, g = acceleration due to gravity, k is a damping coefficient and v = velocity of magnet.

This differential equation's solution is v(t) = (mg/k).[1 - exp(-kt/m)] (if v = 0 when t = 0).

This would mean that the magnet quickly approaches a terminal velocity and then moves at an almost constant speed, asymptotically approaching this terminal velocity.

This appears to match what happens when the magnet is dropped in real life, so does that differential equation have any merit? (It implies that the force opposing the magnet's motion would be proportional to its velocity, would this be a reasonable approximation?)

Also, in which direction do the eddy currents flow in the copper tube? Do they flow around its circumference, or do they flow parallel to the magnet's fall?

### Re: Magnetic damping

That equation is fine: Faraday's law is the only physics that goes in to give qualitative behaviour. The only trouble is trying to make it quantitative. this equation uses an empirical constant k that has to be determined for every magnet, pipe material and pipe geometry. To calculate that constant, you need to make a simple geometric model of the induced current circuit, give it plausible values and look up conductivity of metal.

That's the tricky bit. If you only want an empirical equation, you're home.

Yes, just Faraday's law.

(Continue next message)

That's the tricky bit. If you only want an empirical equation, you're home.

This would mean that the magnet quickly approaches a terminal velocity and then moves at an almost constant speed, asymptotically approaching this terminal velocity.

Yes, just Faraday's law.

(Continue next message)

### Re: Magnetic damping

This appears to match what happens when the magnet is dropped in real life, so does that differential equation have any merit? (It implies that the force opposing the magnet's motion would be proportional to its velocity, would this be a reasonable approximation?)

Also, in which direction do the eddy currents flow in the copper tube? Do they flow around its circumference, or do they flow parallel to the magnet's fall?

Assuming the axis of the magnet is parallel to the tube, it's easy to demonstrate that they are circumferential: If a slit is cut down the side of the wall, there's no circumferential current so very little eddy current and the magnet falls quickly. A current just below and just above the magnet make an electromagnet countering the permanent one, and falling with it. The distribution of this current is what is hard to calculate.

But if an empirical story is all you want, you're home.

Joe

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