Lenz' Law - request for formula (for 2 given cases)

[Student149]

Homework Statement

Assume a uniform hollow cylindrical copper tube with the following parameters:

1. Inner radius = R₀
2. Outer Radius = R₁
3. Length = L₁ (Length > Outer Radius)
4. Mass = M₁

Now assume a cylindrical permanent magnet with the parameters:

1. Length = L₀ (L₀ < L₁)
2. Mass = M₀
3. Magnetic Strength = G₀ Gauss (I am unsure how the strength should be measured, so we can change this into any other appropriate parameter as desired.

There are two scenarios we need to consider:

Case 1:

Assuming the copper tube is aligned parallel to y axis, and at rest w.r.t. to an outside observer. The magnet is traveling at velocity V₀ from parallel to y-axis towards the tube at rest. The center of the tube and the magnet are aligned so the magnet would pass through the copper cylinder without touching its inner walls. Both magnet and tube have their length parallel to y axis.

Due to the Lenz's Effect as the magnet passes though the copper tube it would generate eddy current that would slow down the magnet and some of the momentum would be transferred to the copper tube along y axis. Thus, as the magnet exits the tube:

1. The velocity of magnet along x-axis = 0 (since magnet is traveling parallel to y axis)
2. The velocity of magnet along y-axis = Vy₁
   
3. The velocity of tube along x-axis = 0 (since magnet is traveling parallel to y axis)
4. The velocity of tube along y-axis = Vy₂

Case 2:

All the parameters described as in Case 1 are same. Assume a cylindrical rod parallel to x-axis at rest w.r.t. an outside observer. The rod has negligible mass mass, so for sake of simplicity assume Mass = 0 and some appropriate length = L₂. At each end of the rod are two equal copper tubes attached, each tube making an angle +θ and -θ with the x axis.

Now, two magnets are traveling with the velocity v₀ with an angle +θ and -θ w.r.t. the x axis. The magnets and tubes are such that their centers are perfectly aligned and they would pass straight through the tubes if they were simple metallic pieces.

Again, due to the Lenz's Effect, as the magnet exits the tube:

1. The velocity of magnet along x-axis = Vx₁
2. The velocity of magnet along y-axis = Vy₁
   
3. The velocity of tube along x-axis = 0 (since the horizontal components of the velocity would cancel each other out due two two tubes at angle +θ and -θ w.r.t. the x axis)
4. The velocity of tube along y-axis = Vy₂

Homework Equations

The Attempt at a Solution

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I am looking for a formula (plug and play) to solve these kind of problems (since there are too many factors at play due to electromagnetism and moving magnets, if not exact, then a good approximation would also do) but am failing to find any.

Frankly I have not much idea for any attempt at solution, thus the query for formula.

P.S. This is not a homework problem but out of curiosity. So I don't have much background to follow hints but just looking for a plug and play formula.

 

[kuruman]

I don't believe there is a plug and play formula given your statement of what you want to do. To avoid numerical integration (which means no formula), I would model the magnet as a point magnetic dipole. Even so, integrating the eddy currents over the length of the cylinder to find the ohmic losses looks like a big chore.

 

[Student149]

I don't believe there is a plug and play formula given your statement of what you want to do. To avoid numerical integration (which means no formula), I would model the magnet as a point magnetic dipole. Even so, integrating the eddy currents over the length of the cylinder to find the ohmic losses looks like a big chore.

Thank you. I understand. I would be fine with any decent approximation of the two cases. Say the final velocity for both the tube and the magnets has an error of + or - Δ for some Δ (of course the Δ being known) ?

 

[Student149]

Can anyone please help with this??

Its fine if integrating over a path is necessary to get the answer (an example would be helpful).