The On-Axis Magnetic Field for a Single, Helical Turn Coil

Science and engineering students in the second semester of a calculus-based physics sequence typically study and measure the on-axis magnetic field for a multiple, circular turn coil. There are four benefits to this approach: 1) an analytical solution is easily obtained, 2) the coil is easily constructed using tightly wound, high-gauge wire where each turn is approximately circular, 3) the magnetic field along the axis of symmetry is parallel to that axis, which simplifies the placement of a magnetic field sensor, and 4) the magnetic field strength can be increased by increasing the number of turns, which increases the signal to “noise” ratio. This approach is fine as an introduction, but it doesn’t address several practical questions. How does one calculate the magnetic field for a current-carrying wire that isn’t in the shape of a line, circle, or multiple concentric circles? Are all coils made of high-gauge wire? Aren’t coils helical? Under what conditions can a circular coil in theory be used to represent a helical coil in practice? How does one measure small magnetic fields in the presence of other magnetic fields? This paper presents a secondary approach that begins to address these questions. It determines the on-axis magnetic field for a single, helical turn coil and then details the setup and procedures used for an experimental confirmation. It is presented to students as a guided experiment.