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

Abstract

Adiabatic invariants are fundamental to plasma physics but are often treated as purely theoretical concepts in undergraduate courses due to the difficulty of experimentally demonstrating them. This paper presents a pedagogical experiment to visualize and quantitatively verify the conservation of the magnetic moment ($μ$) and the longitudinal invariant ($J$) using a standard educational electron charge-to-mass ratio apparatus configured as a magnetic bottle. By analyzing long-exposure photographs of the electron beam trajectory, we reconstructed the helical motion and calculated the invariants under different magnetic field configurations. Our results verify the conservation of the longitudinal invariant ($J$) with a ratio of 0.98 between configurations. The magnetic moment ($μ$) exhibited a coefficient of variation of approximately 7\%, a deviation consistent with the presence of collisional effects in the tube. These findings demonstrate that complex plasma dynamics can be effectively studied using accessible laboratory equipment, providing a valuable bridge between theory and experiment for physics students.

Figures (7)

• Figure 1: A schematic representation of the non-uniform magnetic field in a magnetic bottle, illustrating the field lines, two coils and the trapped particle (shown in red). The magnetic field is stronger in the regions near the coils, where the field lines get narrowed.
• Figure 2: The schematic of the experimental setup depicts: (a) the large coils (from PASCO SE-962), (b) the small coils (homemade), (c) the iron cores, (d) the base for the charge-to-mass setup (from PASCO SE-962), (e) the helium gas bulb, (f) the acrylic sheet, (g) miscellaneous supports for securing the small coils, and (h) a laboratory jack.
• Figure 3: Electron Beam Trajectory Divided into Segments. The figure shows (a) the top view and (b) the side view of the electron beam, where its trajectory was divided into three segments for observation. Segment 1 follows the trajectory from the trigger tip to the first bounce, Segment 2 follows the trajectory from the first bounce to the height of the trigger tip, and Segment 3 follows the trajectory from this latter point to the point of the second bounce.
• Figure 4: FEMM simulation scheme of the magnetic flux density for the coil configuration. The left panel shows the magnetic flux density plot, where the magnetic flux lines are represented by black curves. Solid blue lines correspond to the boundary of the small coils, while dashed blue lines indicate the boundary of the large coils. The right panel contains the legend (color scale), which correlates the magnetic flux density magnitude with the colors in the left image.
• Figure 5: Electron beam trajectory graph for the top view of Segment $1$ in Configuration $1$. The blue curve represents the original data, meanwhile the orange curve represents the data after the interpolation and the Gaussian filter.