A free electron, by definition, is not a standing wave. Instead, it is a particle that moves through space and carries electric charge. A standing wave, on the other hand, is a pattern of oscillation that appears to be stationary and does not propagate through space.
Standing waves can occur in electromagnetic fields under certain conditions. However, they typically require a source of excitation and boundaries or constraints that define the region where the standing wave can form. These boundaries can be created by reflective surfaces, resonant cavities, or interference effects.
In the case of an electromagnetic field, standing waves can be generated in structures like waveguides or resonant cavities. A waveguide is a hollow metal structure that confines and guides the propagation of electromagnetic waves. By properly designing the dimensions of the waveguide, it is possible to create standing wave patterns at specific frequencies.
Resonant cavities are another example where standing waves can be formed. These are enclosed regions bounded by reflective surfaces that allow certain frequencies to resonate and form standing waves within the cavity. The resonant frequencies are determined by the dimensions and geometry of the cavity.
In both cases, these standing waves are sustained by a constant excitation source, such as an oscillating current or voltage applied to the system. The boundaries or reflective surfaces of the structure help create the necessary conditions for the formation of standing waves by reflecting and interfering with the incident and reflected waves.
Without a source of constant excitation and boundaries or constraints, it is not possible to generate standing waves in an electromagnetic field. The presence of these factors is crucial to establishing the necessary conditions for standing wave formation.