How does the temperature of a copper pipe affect the time it takes a magnet to fall through it?

One of my favourite topics of the Physics HL syllabus is topic 11 - electromagnetic induction. Diving deep into topics that are closely related to my everyday life was extremely interesting, such as understanding how electric current can be generated with a revolving coil, or how an electric motor or transformer works. All these concepts are at the basis of our existence: our life would most definitely be very different without them. It felt fascinating to me how energy could be created with no visible contact between components, as an interaction involving invisible forces .

 

While learning about Lenz’s law and electromagnetic induction, I found the “falling magnet inside a copper tube” experiment particularly interesting. It appears as if the magnet defies the laws of physics by falling slower than expected, as if calmly levitating towards the ground, supported by a hidden force. After having studied the precise explanation and theory behind it, I realised how this phenomenon can be very helpful in real world applications, for example to decelerate a fast moving object without the use of friction, such as maglev trains, greatly reducing wear on components. I also began thinking about ways to maximise the decelerating effect on the magnet, and one aspect that could impact the performance of the system came to mind: temperature.

The central research question of this paper is as follows-

 

How does the temperature of a copper pipe affect the time it takes a magnet to fall through it?

When a magnet falls through a copper pipe, the moving magnetic field creates a change in magnetic flux in the pipe, ΔΦb. According to Faraday’s law of induction, when a change in magnetic flux is present, an electromotive force, ε is induced in the pipe. As shown by Neumann’s equation , the induced electromotive force is directly proportional to the rate of change of the magnetic flux ΔΦb -

 

\(ε = − \frac{ΔΦ_{B}} {Δt}\)

 

The minus sign is due to Lenz’s law , stating that the direction of the induced current is such as to oppose the change that created the current. This means that the induced current will produce a magnetic field that is opposing the magnetic field of the falling magnet. Put simply, this is why the magnet falls slower in the copper tube when compared to a fall outside of the pipe. However, there are some more considerations to make and other physical phenomena to take into account.

 

The magnetic flux Φb is the number of perpendicular magnetic lines passing through a surface . It can be expressed as -

 

Φb = B ⋅ A ⋅ cosθ

 

With B being the magnetic field strength in tesla (T), A the surface area and θ the angle between the lines and the normal to the surface, n.

However, as seen in figure 2, the surface of a copper tube is not flat, thus the definition of magnetic flux previously given and shown in figure 1 cannot be used in this case. Neumann’s equation cannot be directly used to accurately calculate the induced emf in the pipe. Searching for papers dealing with this specific situation online, on different databases such as Google Scholar or Jstor resulted in very complex literature, which took into consideration the wall thickness with complex formulas which I wasn’t able to understand. Instead, I found an online video which explained the physical phenomenon I wanted to investigate, through Eddy currents and Neumann’s equation. Only the relevant formulas from the analysis will be reported, for the complete derivation of the formulas refer to the video cited.

As seen in figure 3, the model assumes that the copper pipe can be represented as an infinite amount of coils, and that the magnet can be represented as a dipole. These are assumptions that are made in order to simplify the formulas, so they don’t allow for a completely correct analysis of the phenomenon. However, they are far more accurate when compared to only using Neumann’s equation. The magnetic field is assumed to be represented by the magnetic field \((\vec{B} )\) produced by a dipole, which equation is -

 

\(\begin{equation} \vec{B} = \frac{\mu_0}{4\pi} \left[ \frac{3\vec{r}(\vec{m} \cdot \vec{r})}{r^5} - \frac{\vec{m}}{r^3} \right] \end{equation} \)

 

Where μ0 is the magnetic permeability of free space (4.7 x 10-7 H ⋅ m-1), \(\vec{r} \)is the position vector of the point experiencing the magnetic field, and \(\vec{m}\) is the magnetic moment of the dipole. The equation is also expressed in the z component, as it will be needed to later calculate the induced electromotive force. The equations comes out as -

 

\(\begin{equation} B_z = \frac{\mu_0 m}{4 \pi} \left( \frac{3z^2}{(\rho^2 + z^2)^{\frac{5}{2}}} - \frac{1}{(\rho^2 + z^2)^{\frac{5}{2}}} \right) \end{equation} \)

 

The same equation will be re-written using cylindrical coordinates, more suitable to a cylindrical tube. The coordinates are z, which is the vertical displacement inside the tube, ρ, the radius of the tube, and φ, the angular distance. The equation is -

 

\(B_{\rho} = \frac{\mu_0 m}{4 \pi} \cdot \frac{3z\rho}{(\rho^2 + z^2)^{\frac{5}{2}}} \)

 

Using the previous two equations, the emf induced within an infinitesimally small ring, part of the tube, can be calculated. This yields -

 

\(\epsilon = \frac{3 \mu_0 m}{2} \cdot \frac{R^2 z}{(R^2 + z^2)^{5/2}} \cdot v \)

 

Where R is the radius of the tube and v is the velocity of the dipole. The direct proportionality between velocity and induced emf is clear, and respects the initial explanation of Neumann’s equation (higher the change in flux, higher the emf). The force dF exerted on the dipole by the ring can be calculated next, utilising all the previous formulas. The formula comes out as -

 

\(dF = \frac{9\mu_0^2 m^2}{8\pi} \cdot v\sigma \cdot \frac{R^3 z^2}{(R^2 + z^2)^2} \cdot dz \cdot dR \)

 

To find the total force exerted on the dipole by the tube this formula needs to be integrated for the dimensions of the pipe. However, it is enough for the purposes of this paper, as it shows the dependence on the conductivity of copper σ.

 

The conductivity of a metal in relation to its temperature can be approximated by the Wiedemann-Franz law, which equation is -

 

\(σ ∝\frac{ 1}{ T}\)

 

With T representing the temperature in Kelvin. However, this proportionality holds only for very low values of T (a few Kelvins), or for high temperatures, as shown in research by Rosenberg. The more accurate formula for the relation between conductivity (or resistivity, its reciprocal) is the Bloch–Grüneisen formula. However, it is a very complex formula and way beyond the scopes of this paper. As a simplification, it still confirms that the resistivity of metals increases with temperature, although the relationship changes for each metal. However, experimental data by Richard Matula shows how this relationship, especially in the temperature range that this paper is concerned with (250- 400K), can be assumed to be linear.

 

Having calculated the force exerted by an infinitesimally small ring on a dipole, and assuming that, after integration, it represents the magnet falling into a tube, it has been shown that the force exerted on the magnet, and the time taken for the magnet to fall through the tube, is directly proportional to the temperature. For this reason, the time taken for the fall decreases with the increase in temperature.

In order to address the question stated in 1.1, an experiment was set up, to measure the correlation between variation in temperature of the pipe and time of the fall.

The independent variable in this experiment is the temperature of the copper pipe, while the dependent variable is the time it takes for the magnet to fall vertically through it. The controlled variables are the dimensions of the tube, the dimensions of the magnet, the temperature of the magnet and the ambient temperature.

The experiment is set up as follows - a vertical 75 cm tube, with a 2 cm insulating sleeve, held by a stand with two clamps, one at the top end of the tube, and the other at the bottom end. Two magnets of magnetic field strength 50 mT are used. A Vernier GoDirect Surface temperature sensor is attached to the stand, the probe inserted 10 cm between the outer surface of the tube and the insulation. The Vernier Graphical Analysis software will be used to record the temperature of the pipe and check that it remains constant throughout the trials. A camera will be placed on a tripod approximately 5 metres away from the stand, in order to completely see both of the openings of the tube. For the same reason, the camera should be placed at half the height of the stand from the ground, or 60 cm. The camera will then be used to record a 60 fps (frames per second) video of the magnet drop. This yields more accurate results when compared to other methods, such as measuring time with a stopwatch (see appendix for complete procedure). As the experiment will measure 4 temperatures, different methods will be used to cool or heat up the pipe, such as a hairdryer or a freezer.

Here follow the steps taken in order to obtain the 5 different values of the time taken for the magnet to fall for each of the 4 temperatures of the copper pipe. For temperature 1, the tube was left overnight in a -25°C freezer, and the trials were conducted when the pipe reached -20°C, outside of the freezer. Temperatures 3 and 4 require the pipe to be heated to 50°C and 100°C respectively using a hairdryer.

• Start recording a video at 60 FPS (frames per second) on the camera.
• Start collecting temperature data in the “Vernier Graphical Analysis” software.
• Manually drop the magnet through the tube, taking care not to hit the sides and to drop it exactly at the opening of the pipe.
• Repeat step 4 until five trials are obtained.
• Repeat steps 4 - 5 for all temperatures.
• Import the 4 videos (one for each temperature) in Adobe Premiere Pro.
• With the playhead, select the exact frame in which the magnet starts to enter the tube and record it.
• Move the playhead to the frame in which the magnet is completely out of the pipe and record it (left and right arrows can be used for precise movement between frames).
• Subtract the final frame count from the initial frame count to obtain the change in frames, i.e. the time it took (in frames) for the magnet to fall, and record it on a spreadsheet, together with the recorded average temperature.

As the experiment deals with both powerful neodymium magnets and high temperatures, caution must be used when conducting it. Protective gloves will be used when handling the hot pipe, along with common sense, in order to avoid burns and bruises. Also, the insulation chosen will be able to withstand such high temperatures without melting or burning. The magnets will always be handled with attention and far from any metal surface or object, in order for them not to be attracted and possibly cause injury. Only one person will be near the stand and conduct the experiment, so as not to create confusion and possible safety hazards.

The uncertainty is ±1 frames, the smallest increment in the measurement system.

The mean and standard deviation for each temperature will be calculated, obtaining a table. An example with the first temperature (-20°C) is given -

 

\( \overline{T_1} = \frac{214 + 215 + 212 + 213 + 212}{5} = 213.2 \text{ frames} \)

 

\(\sigma_1 = \sqrt{\frac{\sum_{i=1}^{5} (x_i - 213.2)^2}{4}} = 1.303840481 \approx 1.3\)

 

All other calculations have been computed in a Google Sheets spreadsheet, to speed up the process and avoid errors. The final table is -

The time taken for the magnet to fall through the tube will now be converted into seconds, in order to have an SI unit to work with. As the videos were recorded at 60 fps, the conversion is as follows -

 

1 s = 60 frames

 

\(1 frame =\frac{ 1} {60} s\)

 

For this reason, it will be sufficient to divide the number of frames by 60 to obtain the time in seconds. An example from trial one of the first temperature (-20°C) is given -

 

Time in frames - 214

 

\(Time \,in \,seconds -\frac{ 214} {60 }= 3.56\)

 

A new table will be presented, with the time in seconds. All the conversions have been done through a Google Sheets spreadsheet, as before.

The processed data will now be plotted on Vernier Logger Pro, showing error bars representing the standard deviation previously calculated and lines of best fit. As the relationship between temperature and fall time should be linear, a linear fit has been plotted, showing how the values for the first three temperatures (-20°C, 21,5°C and 50C) are consistent with each other, while the 100°C measurement can be considered an outlier, for reasons that will be discussed in the following section.