In a region free of currents, magnetostatics can be described by the Laplace equation of a scalar magnetic potential, and one can apply the same methods commonly used in electrostatics. Here we show how to calculate the general vector field inside a real (finite) solenoid, using only the magnitude of the field along the symmetry axis. Our method does not require integration or knowledge of the current distribution, and is presented through practical examples, including a non-uniform finite solenoid used to produce cold atomic beams via laser cooling. These examples allow educators to discuss the non-trivial calculation of fields off-axis using concepts familiar to most students, while offering the opportunity to introduce important advancements of current modern research.
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