Magnet in a copper tube: eddy-current braking versus 'copper is not magnetic'
A student drops a small cylindrical neodymium magnet through a vertical copper tube. The tube is 0.60 m long, with inner diameter 12 mm and wall thickness 1.5 mm. The magnet is 10 mm in diameter, 15 mm long and has mass 8.2 g. Two light gates, 0.15 m apart, record speeds 0.65 m/s and 0.66 m/s near the bottom of the tube; the total fall time is 0.92 s. The student then cuts a narrow slit along the full length of a second identical tube and repeats the drop; the fall time is 0.31 s. The magnet is slightly tilted as it falls, and the tube is not perfectly vertical. The student claims the slit tube shows that 'copper is not magnetic' and that the slowing in the uncut tube is due to air resistance.
1. Use the two light-gate speeds to decide whether the magnet is at terminal velocity near the bottom. Calculate the magnetic drag force at that point and compare it with the weight.
Read the worked answer
Weight: mg = 0.0082 × 9.81 = 0.0804 N.
Acceleration over the 0.15 m between gates: a = (v₂² − v₁²)/(2s) = (0.66² − 0.65²)/(2 × 0.15) = (0.4356 − 0.4225)/0.30 = 0.0437 m/s² downward.
Net downward force = ma = 0.0082 × 0.0437 = 3.58 × 10⁻⁴ N. The magnetic drag acts upward, so drag = mg − ma = 0.0804 − 0.00036 = 0.0800 N.
The drag is almost exactly equal to the weight, and the acceleration is only about 0.4% of g. The magnet is very close to terminal velocity; the small positive acceleration means the true terminal speed is slightly above 0.66 m/s. Air resistance at 0.65 m/s is orders of magnitude smaller than 0.080 N, so it cannot be the main drag.
2. Explain, using Faraday's law and Lenz's law, why the slit increases the fall time ratio and why the student's 'copper is not magnetic' and 'air resistance' claims are wrong. Estimate the average speed in each tube and comment.
Read the worked answer
Average speed in the uncut tube = 0.60/0.92 = 0.652 m/s. In the slit tube = 0.60/0.31 = 1.94 m/s. The slit tube is about three times faster on average, so the slowing is not a small effect.
As the magnet falls, the magnetic flux through each horizontal loop of the copper tube changes. Faraday's law gives an induced emf, and because copper is a conductor, eddy currents flow. By Lenz's law these currents produce a magnetic field that opposes the change in flux, so the force on the magnet is upward and opposes the fall. This is electromagnetic braking.
Copper is not ferromagnetic, so the student's phrase 'not magnetic' is misleading: a changing flux can still induce currents in any conductor. The slit interrupts the large circular eddy-current paths, increasing the effective resistance. For the same induced emf, the currents are smaller, so the braking force is smaller and the magnet falls faster. Air resistance cannot explain the uncut tube because the drag needed is about 0.080 N, whereas air drag on a small magnet at 0.65 m/s is negligible.
3. The student cools the uncut tube with liquid nitrogen (77 K) and drops the magnet again. Predict and justify whether the average speed will be lower or higher than at room temperature. Discuss one factor that could make the prediction unreliable.
Read the worked answer
The average speed should be lower at 77 K. Copper's resistivity falls strongly as temperature decreases, so for the same induced emf the eddy currents are larger. Larger currents give a larger magnetic braking force at the same speed, so the magnet reaches a lower terminal speed and takes longer to fall.
A factor that could make the prediction unreliable is the thermal contraction of the tube. The bore becomes slightly smaller, and the magnet may tilt or rub, adding mechanical friction that is not electromagnetic. Condensation or ice on the tube could also affect the motion. In addition, if the magnet's orientation changes, the flux pattern and eddy-current distribution change, so the braking force is not simply a function of resistivity.