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 ($\mu$) 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 ($\mu$) 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.