By Odin
Electromagnetic induction caused by a magnet free-falling through wire coils in series is used to measure acceleration due to gravity. Through the capture and analysis of more than 100 magnet drops, graphs of induced voltage over time are analysed to provide measurements of the magnet’s speed and hence acceleration as it falls through successive coils.
The results achieved (when analysed) generate a value of (g \approx 9.77\ \text{m s}^{-2}). This result is discussed and the impact of resistive forces (air resistance, friction and electromagnetic), especially at higher speeds, is considered.
Electromagnetic induction is introduced to many Physics students because it is covered across most GCSE and A-level syllabi. A basic explanation is:
“A voltage is induced in a conductor or a coil when it moves through a magnetic field or when a magnetic field changes through it.”
For a current to flow, a potential difference must be present. A potential difference can be induced in a conductor when relative movement between the conductor and the magnetic field exists. To give a practical example: imagine two dipole magnets placed next to each other so that a uniform magnetic field is formed between the south pole of one and the north pole of the other.
Fig. 1: The passing of a conductive wire between two dipole magnets, facing each other so that they create a uniform magnetic field.
By moving a wire of conducting material between these two magnets so that the wire cuts the magnetic field lines normally (Fig. 1), a voltage is induced. This voltage is tiny, and to create a voltage of a larger magnitude we can:
- manipulate the wire into a coil to increase the number of magnetic field cuts per unit time,
- use a stronger magnet (more frequent/denser field lines),
- move the magnet through the wires faster.
By connecting a voltmeter to a coil of wire and dropping a dipole magnet through it, we can analyse the characteristics of the induced voltage at specific positions along the coil using a voltage–time graph.
Fig. 2a: The falling magnet approaching the coil and the anti-clockwise current within.
Fig. 2b: The falling magnet departing the coil and the clockwise current within.
Fig. 3: A graph illustrating induced voltage through a coil of wire by a falling magnet, labelled with the magnet’s positions throughout.
Immediately, the graph presents two distinct features: a peak and a trough. As the magnet falls into the coil, the magnet’s field creates a change in magnetic flux linkage within the coil. By Faraday’s law of electromagnetic induction, this results in a positive induced voltage (the initial peak). As the magnet exits the coil there is again a change in flux linkage, but the induced voltage is negative (the trough).
Equation 1 (Faraday’s law): “Voltage is equal to the negative rate of change in magnetic flux linkage with respect to time.”
As the magnet falls into the coil, the induced voltage causes an induced current to flow anti-clockwise (Fig. 2a). This results in a magnetic north pole developing at the top of the coil and a south pole developing at the bottom. The direction of current (and thus magnetic polarisation) is determined by Lenz’s law, which states:
“The direction of the induced EMF and the resulting current in a closed loop is such that it opposes the change in magnetic flux that produced it.”
Therefore, as the magnet approaches the top of the coil the induced current is anti-clockwise such that it creates a magnetic north pole which repels the incoming magnet. As the magnet exits the coil the induced current reverses to clockwise (Fig. 2b), the poles of the coil reverse, and it exerts an upward attractive force on the magnet—again opposing the change in flux. This reversal of induced current means the induced voltage changes sign: on entry it is positive, and on exit it is negative.
Another feature of the graph (Fig. 3) is the instant when the induced voltage is equal to zero. This occurs when the middle of the magnet is inside the coil, because the magnetic field around the middle of the magnet is relatively uniform: at that instant there is no change in flux linkage and therefore the induced voltage is zero (Faraday’s law).
A small but significant detail is that the magnitude of the trough is greater than the magnitude of the peak. As the magnet falls, it accelerates, so by the time it exits the coil it is travelling faster than when it entered. This is also seen in the width: the trough is narrower, implying it takes less time to exit than to enter. Therefore the rate of change of flux linkage is greater on exit, producing a larger induced voltage. In Faraday’s law terms, for the same (\Delta(N\Phi)), a smaller (\Delta t) gives a larger (|V|).
This raises the question of whether the magnet is accelerating at (g \approx 9.81\ \text{m s}^{-2}). At the start ((t=0)), acceleration is (g). As the magnet approaches a coil, the top of the coil becomes a north pole and repels the magnet: an upward force means the acceleration is less than (g). On exit, an upward attractive force again reduces the acceleration. However, when the magnet is in the middle of the coil (induced voltage zero), there is no induced current and therefore no coil-generated magnetic field; at that instant the only force is weight, so the acceleration is (g).
By linking induced voltage to the magnet’s position, a voltage–time graph can be treated as a proxy for a position–time description, allowing determination of speed between two points. To measure acceleration, multiple speed readings are needed to compute changes in speed over time.
Hypothesis: By analysing the voltage–time graph of a magnet falling through a series of coils, we can approximate the acceleration due to gravity ((g)). By correlating the induced voltage to the magnet's position and speed, and calculating the change in speed over time, we can estimate the magnet's acceleration and use it to determine an approximation for (g).
The hypothesis tested is that the induced voltage caused by a magnet falling under gravity through coils in series can be used to measure the magnet’s position at different times, and hence acceleration due to gravity.
Apparatus (see Fig. 4–7):
- A Perspex tube of approximately 1 metre in length (≈101.5 cm) and 3 cm diameter, with a 2.4 cm bore.
- The tube held vertically using clamp stands, G-clamped to the desk.
- Seven 50-turn copper coils (≈2 cm height), evenly spaced at approximately 15 cm intervals around the tube.
- The seven coils wired in series using one continuous length of copper wire.
- A PASCO SPARKvue voltmeter attached to the ends of the coil wire using crocodile clips (after stripping enamel).
- Datalogging software polled voltage at 2000 samples per second, via Bluetooth.
- The software started recording a run when the detected voltage exceeded 4 mV (0.004 V) (signifying the magnet approaching the first coil).
- Projectile: a cylindrical neodymium magnet (1 cm height, 2 cm diameter) attached to wooden dowelling (11 cm length, 2 cm diameter), total mass 86.3 g.
- Purpose of dowelling: prevent rotation, which could introduce non-uniformities in the magnetic field and reduce accuracy.
- Before release: the projectile was aligned magnet-first at the top of the tube, protruding ≈1 cm into the tube so it fell cleanly without striking the wall.
- Tube bore 2.4 cm vs projectile diameter 2.0 cm → clearance ≈0.2 cm around the projectile.
- Catch: a rag held by clamp stands to prevent damage when the projectile exited.
- Recording ended after 0.5 s (long enough for the magnet to pass through and for induced voltages to cease).
Variables:
- Control variables
- Magnetic field strength of the magnet
- Number of turns in the coils
- Number of coils
- Distance between coils
- Independent variable
- Time
- Dependent variable
- Induced voltage
The projectile took multiple forms during early runs.
- Cuboid neodymium magnets (0.5 × 1 × 2.5 cm) were tested first. Although they fit down the tube, they produced unsatisfactory data: instead of the poles being on the top and bottom (0.5 × 1 cm faces), the poles were on the 1 × 2.5 cm side faces. This led to erratic voltage–time data because the magnetic field lines were not being cut in the intended orientation.
- Fig. 8: Illustration of cuboidal magnet.
- Fig. 9: Graph produced by cuboidal Nd magnet.
- Next, a pair of cylindrical magnets (with poles on the top and bottom circular faces) were tried. Stacking two magnets increased the cylinder length and helped reduce rotation. However, combining two magnetic fields sometimes produced a slightly non-uniform field, leading to voltage–time curves that were more angular than expected.
- Fig. 10: Double cylindrical magnets attracted to each other.
- Fig. 11: Example voltage–time graph from dropping two cylindrical magnets.
- Finally, a single cylindrical magnet with diameter matching the tube bore was obtained and attached to wooden dowelling to prevent rotation. This configuration was used for data collection runs (118 total). The single-poled magnet plus anti-rotational dowelling produced very satisfactory voltage–time graphs.
- Fig. 12: Final form of the projectile.
- Fig. 13: Example “satisfactory” voltage–time graph.
Working independently, the drop was repeated 117 times with the improved projectile, for 118 runs in total.
The SPARKvue datalogging software allowed export of each run to .csv. The dataset was imported into a Python Jupyter Notebook, enabling segmented execution (no need to re-import data when adjusting plotting instructions) and convenient graph display.
Matplotlib was used to generate plots, including an overlay plot of all runs to help spot outliers.
- Fig. 14: Screenshot of the Python Notebook within the integrated development environment.
- Fig. 15: A graph overlaying each run within the dataset of 118 drops.
After completing data collection, equipment was tidied and returned to the Physics Department storeroom.
All code and datasets are hosted on the author’s GitHub page.
After importing the data and excluding invalid runs, analysis proceeded using the voltage–time data. To find speeds down the tube, the key times required are when the magnet is between coils and the induced voltage is equal to zero.
A Python function was written that takes two equally sized arrays of:
- x-values (time; independent variable)
- y-values (induced voltage; dependent variable)
The function iterates through the y-values looking for a sign change. When a sign change is detected, it calculates the exact x-coordinate of the zero crossing using linear interpolation. If a y-value is exactly zero, it is also added to the list of crossings.
Edge cases were handled:
- Multiple zero crossings within a very small interval (< 0.01 s) were not added consecutively (to avoid noise).
- The function searched backwards from the end to find the last significant crossing where y < -0.005, to capture the end of induced voltage accurately; any subsequent noise was rejected.
Passing each run through the zero-crossing function before plotting allowed the coordinates to be visualised and checked before calculation.
- Fig. 16: Plot of the first run with zero-crossing points visualised.
To measure speed at points between coils, half of the zero crossings were removed (those corresponding to the magnet being within a coil). This was done by redefining the list from the second coordinate and taking every second coordinate thereafter (shown in Fig. 16 as red crosses).
From the apparatus design, the gap between the tops of successive coils is (0.15\ \text{m}). Speed between two midpoints can be found from the time difference between successive “between-coil” zero-crossing points:
Equation 2 (speed):
Acceleration between adjacent midpoint locations is then found by the change in speed over the corresponding time interval:
Equation 3 (acceleration):
This produced an array of five acceleration values, measured throughout the last five sectors between coil midpoints (the red crosses in Fig. 16). Repeating the velocity and acceleration calculations for all 118 runs enabled creation of box-and-whisker plots showing the range of accelerations in each sector.
An annotated photo of the tube was also used to show which sector corresponds to which physical location (with the red/yellow crosses from Fig. 16 included for clarity).
- Fig. 17: Box-and-whisker plots of accelerations in sectors 1–5.
- Fig. 18: Tube labelled with sectors.
Sector 5 was extremely uncertain and anomalous, so it was removed from the final determination of (g). For sectors 1–4:
- (9.81\ \text{m s}^{-2}) lies within each range of accelerations,
- the median of the means is close to 9.81.
By averaging all obtained accelerations across sectors 1–4, the mean acceleration was:
[ g \approx 9.77\ \text{m s}^{-2} ]
This is slightly below the accepted value by (0.04\ \text{m s}^{-2}); reasons are discussed in the conclusion.
The obtained value (9.77\ \text{m s}^{-2}) is fairly accurate, being about 0.41% below the accepted value (9.81\ \text{m s}^{-2}). This raises the question of which factors contributed to the magnet accelerating towards the ground at a slower value than expected.
Newton’s second law states the resultant force is mass times acceleration.
Equation 4 (Newton’s second law):
With the projectile mass constant at 86.3 g, the average resultant force during trials is (0.834\ \text{N}). Under acceleration (g), the resultant force would be (0.847\ \text{N}). The difference suggests a total resistive force of approximately:
[ F_{\text{resist}} \approx 0.007\ \text{N} ]
-
Air resistance
The measured (g) assumes no opposing forces. To reduce air resistance, the experiment would need to be performed in a vacuum chamber (not available). A vacuum setup would also require a remote release mechanism (e.g., an electromagnet), and would be far more time-consuming between drops (depressurise → reset magnet → repressurise). -
Friction / collisions with tube walls
With a bore of 2.4 cm and projectile diameter 2.0 cm, the clearance is only ~0.2 cm around the projectile. Any lateral force during release could lead to collisions with the tube wall. Such collisions create reaction forces and can cause repeated bouncing/rubbing, increasing friction and reducing acceleration. -
Electromagnetic forces from induction
As described in the introduction, the magnet induces currents in coils, which generate magnetic fields opposing the magnet’s motion (Lenz’s law). On approach, the coil’s top becomes a north pole, repelling the magnet; on exit, the induced poles reverse, producing an upward attractive force. Both effects reduce downward acceleration.
A major discordance was the stark decrease in acceleration through the seventh (last) coil and out of the tube. The data produced a median acceleration of about (-1.99\ \text{m s}^{-2}) for sector 5 (Fig. 17). One proposed explanation is that, by coil seven, the induced magnetic field becomes strong enough that the upward electromagnetic force can exceed the weight of the magnet—yielding an upward resultant force (negative acceleration). This did not occur in all trials, but the force appears consistently present, as the upper limit for sector 5 was less than half of (g).
A positive, linear relationship between the magnet’s speed and induced voltage is captured by another form of Faraday’s law:
Equation 5 (Faraday’s law, speed form):
If (N) (turns) and (B) (field strength) are constant, increasing velocity (v) increases induced voltage (V), and thus increases the induced magnetic field strength—creating a greater resistive force against the magnet.
All data tables in the appendix have headings and units, and the exact conditions for data being taken are clear. An exception might be tables with concluding values from different aspects of the investigation.
Including a table of all recordings and results would be extensive, so the Python program, charts, and entire dataset are instead hosted on GitHub.
Figure 15 (repeated in the original document) displays a summary of all voltage–time data obtained.