# Introduction

Faraday’s law predicts how magnetic fields create an electromagnetic force by interacting with a circuit. Lenz’s law shares the same fundamental formula as Faraday’s but further clarifies that the induced current acts opposite the initial magnetic field.

A common experiment to showcase Lenz’s law is this:

Acquire a neodymium magnet and a tube made of a conductive non-magnetic metal such as copper. When the magnet is dropped down the tube, it falls much slower than it would if there were no tube. The falling magnet induces an electromotive force (EMF), also known as an eddy current, within the tube. This force increases in strength with better electrical conductivity and opposes the force of gravity on the magnet. This is the experiment I’ve done for this post.

In essence, Lenz’s law is a manifestation of the Law of Conservation of Energy. Given that the magnetic flux of the system is changing, the energy of the system changes as well. This increase in energy must be released somehow, and Lenz’s law is the result. The induced force must act opposite to the initial magnetic field. Otherwise, the induced EMF would accelerate the magnet, which would, in turn, increase the change in flux and create a chain reaction of acceleration. This would make it possible to create infinite energy from practically nothing, so let’s be glad that isn’t happening.

For my project, I’ll be using a copper tube, an aluminum tube, and a plastic tube (as a control). The copper tube is 104.5 cm long, 1 mm thick, and has an inner diameter of 10.9 mm. The aluminum tube is 104.5 cm long, 1.3mm thick, and has an inner diameter of 10.1 mm. Lastly, the plastic tube is 92.5 cm long with an inner diameter of 12.25 mm. For reference, the electrical conductivity of copper is about 5.8 x 10⁷ S/m, and aluminum’s is about 3.8 x 10⁷ S/m. Given tubes with identical dimensions, this would mean that the copper tube opposes the magnet’s fall more. However, my tubes do not have the same dimensions, which I will discuss more later on.

The neodymium magnet I’ll be using is a 10 mm by 10 mm cylinder with a mass of 6 g. My dad somehow had a Gauss meter lying around, so we measured the flux density to be around 5,200 G at the surface and around 2,000 G at a distance of 0.5 mm.

# The Experiment

Alright, let’s get into the experiment. For the procedure, I simultaneously dropped the magnet in a tube and started a stopwatch. As soon as the magnet exited the other end, I stopped the stopwatch. The following results are the averages over five runs each, with .2 seconds subtracted to account for reaction time.

Aluminum tube: 20.95 s | Copper tube: 21.56 s | Plastic tube: 0.45 s The magnet descending through the copper tube, illuminated by a flashlight

You may recall that my tubes all have different dimensions. Let’s discuss the discrepancy between the metal tubes first. There are some big complicated formulas that have been created to accurately model the strength of the eddy currents induced in the metal, which I will not be using. The key takeaway from these sizeable formulas is that both the thickness of the metal and the distance from the magnet to the inner wall affect the strength of the eddy currents. Specifically, thinner metal and a larger gap will weaken the currents. A big complicated formula, where δ is the thickness of the metal¹

If the tubes were of identical dimensions, the only thing affecting the strength of the eddy currents would be the conductivity of the metals. Thus, the copper tube would have stronger currents and produce longer fall times than the aluminum tube. However, since the copper tube is wider and thinner, the effect of the eddy currents is diminished. Despite this, the recorded times for the copper tube are still larger, which gets the point across nonetheless.

We still need to account for the discrepancy in the length between the plastic tube and metal tubes, so let’s do some math.

We’re looking for the time it would take for the magnet to fall the additional 12 cm to match the length of the metal tubes for an accurate comparison. To do this, we first need to find what the final velocity of the magnet is after falling 92.5 cm:

Using this velocity as the initial, the time it would take to fall the extra 12 cm is:

Therefore, the time for the magnet to fall 104.5 cm through the plastic tube is 0.47 s. The time for the magnet to fall through 104.5 cm of each tube is as follows (aluminum and copper tubes remain unadjusted):

Aluminum tube: 20.95 s | Copper tube: 21.56 s | Plastic tube: 0.47 s

You may be wondering: “Well, if the magnet in the plastic tube is in freefall, why couldn’t he just calculate the time solely mathematically instead of doing it experimentally?” First, this somewhat helped to standardize the slight tube friction, air drag, and any other environmental biases between the three tests. Second, it allowed me to do more math than would have been necessary, and I like how the math looks in these posts.²

# Discussion and Conclusion

Lenz’s law has many real-world applications like electric motors, card readers, and even the breaking systems for trains. This post was just a little taste of what can be done with Lenz’s law. In my next post, I’ll be predicting and experimentally verifying the eddy current-inflicted drag force on the magnet at terminal velocity, so be on the lookout for that one.

Footnotes
1. Ma, Der-Ming (2011), “The design of eddy-current magnet brakes
2. For those who are wondering, the calculated time using 104.5 for d and 0 for vi is about 0.46 s, so using the experimental results adds around .01 s.

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