Experimental verification of the conservation of the magnetic moment and the longitudinal invariant

This paper presents a pedagogical experiment using a modified electron charge-to-mass ratio apparatus to quantitatively verify the conservation of the magnetic moment and longitudinal invariant in a magnetic bottle, successfully bridging theoretical plasma physics concepts with accessible laboratory practice.

The Gist

Imagine you are trying to catch a slippery fish in a river, but instead of a net, you are using a pair of invisible, magnetic hands. This is the basic idea behind a "magnetic bottle," a device used to trap charged particles like electrons.

This paper describes a classroom experiment where students built a magnetic bottle to test two fundamental rules of physics that usually only exist in textbooks. The goal was to see if these rules hold up when you actually try to measure them with real equipment.

Here is a simple breakdown of what they did and what they found:

The Setup: A Magnetic Trap

Think of the magnetic bottle as a hallway with two heavy doors at either end that are slightly "sticky."

• The Hallway: In the middle, the magnetic field is weak, so the electrons (our "fish") can zip around freely.
• The Sticky Doors: As the electrons move toward the ends, the magnetic field gets stronger. This acts like a mirror. When the electrons hit this strong field, they bounce back, just like a ball hitting a wall.
• The Motion: The electrons don't just bounce back and forth in a straight line; they spiral like a corkscrew as they travel.

The Two Rules They Tested

The scientists wanted to check if two specific "conservation laws" (rules that say certain things must stay the same) were true in their experiment.

1. The Magnetic Moment (The "Spin" Rule)

• The Analogy: Imagine a figure skater spinning. If they pull their arms in, they spin faster. In this experiment, as the electron moves into the "sticky" magnetic field, its sideways spinning speed changes to keep a specific balance.
• The Test: They measured the electron's spin speed at different points in the bottle.
• The Result: The rule held up mostly, but not perfectly. The numbers varied by about 7%.
• Why? The paper explains that the electrons were bumping into gas molecules inside the tube (like a crowded dance floor). These tiny collisions messed up the perfect spin, causing the slight variation. It wasn't a failure of the rule, but a sign that the real world is messier than the perfect math models.

2. The Longitudinal Invariant (The "Bounce" Rule)

• The Analogy: Imagine a pendulum swinging back and forth. Even if you change the length of the string slightly, the time it takes to swing from one side to the other stays surprisingly consistent. This rule says that no matter how the magnetic field changes, the electron will always return to the same "bounce points."
• The Test: They ran the experiment twice with slightly different magnetic field strengths and measured the distance the electrons traveled between their bounces.
• The Result: This rule worked almost perfectly. The two measurements were 98% identical.
• Why? Because this rule looks at the "big picture" of the motion (the whole trip from one end to the other), it is less sensitive to the tiny, messy collisions that happened along the way.

How They Did It

Instead of using expensive, high-tech satellite data, the team used a standard university physics kit (usually used to measure the charge of an electron) and added some extra coils to create the magnetic bottle.

• The Camera Trick: They took long-exposure photographs (like leaving the camera shutter open for 10 seconds) in a dark room. This turned the fast-moving, invisible electron beam into a glowing, visible line on the photo, allowing them to trace its path.
• The Computer Work: They used software to turn those photos into data points, calculated the speeds, and compared them against computer simulations of the magnetic field.

The Bottom Line

The paper concludes that you don't need a multi-million dollar lab to study complex plasma physics. By using accessible equipment, students can actually see and measure these invisible forces.

The experiment proved that:

1. The "Bounce" rule is very robust and holds true even with experimental errors.
2. The "Spin" rule works well, but small deviations (caused by collisions) are normal and expected in the real world.

Ultimately, this experiment bridges the gap between abstract math on a chalkboard and the messy, fascinating reality of how particles actually behave.

Technical Summary

Technical Summary: Experimental Verification of the Conservation of the Magnetic Moment and the Longitudinal Invariant

Problem Statement
Adiabatic invariants, specifically the magnetic moment ($\mu$) and the longitudinal invariant ($J$), are fundamental concepts in plasma physics. However, in undergraduate education, these are frequently treated as purely theoretical constructs due to the difficulty of experimentally demonstrating them. Existing experimental validations often rely on satellite data or complex theoretical modeling, requiring setups inaccessible to standard physics laboratories. This paper addresses the need for a pedagogical experiment that allows students to visualize and quantitatively verify these invariants using readily available educational equipment.

Methodology
The authors designed an experiment utilizing a magnetic bottle configuration constructed from standard laboratory components:

• Apparatus: A helium-filled electron charge-to-mass ratio tube (PASCO SE-9659) was placed between two pairs of coils and iron cores. The setup included a PASCO electron charge-to-mass apparatus coil pair (large coils) and a set of custom-built smaller coils to create a non-uniform magnetic field gradient.
• Configuration: Two distinct magnetic field configurations were established by varying the currents in the large and small coils and the acceleration potential, while maintaining a constant beam injection angle.
• Data Acquisition: Electron beam trajectories were captured using long-exposure photography (6–10 seconds) from two orthogonal perspectives (top and side views) to reconstruct the 3D helical motion. Fluorescent rulers provided spatial calibration.
• Simulation and Analysis: The magnetic field distribution was modeled using Finite Element Method Magnetics (FEMM) via the PyFEMM library. Trajectory data extracted from images (using Inkscape) were converted to physical coordinates. Velocities were calculated by differentiating the path parameter $s$ with respect to time, utilizing energy conservation to determine total velocity. The data were smoothed using Gaussian filters and fitted with polynomial and exponential functions to reduce noise and account for the relationship between parallel and perpendicular velocity components near mirror points.

Key Results
The study calculated the adiabatic invariants for two different magnetic field configurations:

• Magnetic Moment ($\mu$): The magnetic moment was evaluated for three trajectory segments within each configuration. The results showed that $\mu$ was not strictly conserved.
  • Configuration 1 Mean: $5.494 \times 10^{-15}$ Nm/T (Coefficient of Variation [CV]: 7.323%).
  • Configuration 2 Mean: $4.104 \times 10^{-15}$ Nm/T (CV: 6.540%).
  • The deviation from strict conservation (CV > 5%) was attributed to two factors: physical collisions between electrons and the background gas (breaking cyclotron phase) and methodological errors such as parallax and optical aberrations in image processing. The error increased progressively from Segment 1 to Segment 3, supporting the collision hypothesis.
• Longitudinal Invariant ($J$): Calculated over the full bounce motion (Segments 2 and 3), the values were $J_1 = 402,285.569$ m²/s and $J_2 = 410,191.790$ m²/s. The ratio $J_1/J_2$ was found to be 0.98. This result indicates that the longitudinal invariant was conserved within experimental uncertainty, despite the local perturbations affecting $\mu$.

Significance and Claims
The paper claims that complex plasma dynamics, specifically the conservation of adiabatic invariants, can be effectively studied using accessible, low-cost laboratory equipment. The primary contributions are:

1. Pedagogical Bridge: The experiment provides a tangible link between theoretical derivations and experimental reality for undergraduate students, allowing them to visualize the magnetic bottle effect and distinguish between cyclotron and bounce motions.
2. Methodological Training: The study introduces students to essential experimental techniques, including image processing for data extraction, numerical integration of real signals, and the use of FEMM simulations to connect theoretical modeling with measurements.
3. Realistic Modeling: The observed deviations in the magnetic moment serve as a practical case study for discussing the limitations of ideal models and the role of collisions in plasma systems.

The authors conclude that while the magnetic moment showed expected deviations due to experimental and physical constraints, the strong conservation of the longitudinal invariant validates the experimental approach. The work demonstrates that with a relatively simple system, these quantities can be characterized with satisfying results, offering a viable alternative to high-cost or purely theoretical instruction in plasma physics.