Use of a varying turn-density coil (VTDC) to generate a constant-gradient magnetic field and to demonstrate the magnetic force on a permanent magnet

Abstract

In this study, we proposed that a constant gradient magnetic field could be generated by the construction of a current-carrying coil possessing a turn-density (number of turns per unit length) that varies linearly with axial position (i.e., a varying turn-density coil, VTDC). A VTDC is easily constructed and could be used to demonstrate how a ferromagnet placed in a magnetic field experiences a net force. Using the Biot–Savart law, we predicted that a ferromagnetic dipole suspended in a VTDC experiences a force proportional to the length of the dipole. We designed and executed an experiment to test our prediction. A coil with a turn-density varying at a rate of 0.11 cm−2 was constructed. Rare-earth magnets were suspended along the centre of the VTDC using a string that was attached to a mass resting on a scale. The scale reading measured the force experienced by the hanging magnet. The procedures were also carried out in a constant turn-density solenoid (control). A direct linear relationship between force and magnet length was observed in the VTDC, whereas no force was detected in the zero-gradient field solenoid (p < 0.000 001). Linear regressions suggested that the observed data matched the predicted values (95% certainty). The magnetic moment of 0.56 ± 0.21 J/T is in reasonable agreement with remnant magnetization for neodymium magnets of ∼1 T. We found that our easily-built VTDC produced a uniform gradient as no significant differences were observed when magnets were hung at different axial positions. Overall, the results were supportive of the theory. Our VTDC could be readily constructed and used in a physics classroom to demonstrate basic principles of electromagnetism.