A charged particle does indeed exert an electric force on itself when it's in motion. This self-interaction force is known as the "self-energy" or "self-force." However, this self-force is typically negligible and is often neglected in many practical situations.
The self-force arises due to the electromagnetic field produced by the moving charged particle interacting with itself. According to classical electrodynamics, the electromagnetic field is generated by a distribution of charges and currents. In the case of a single charged particle, the field it produces interacts with itself, resulting in a self-force.
However, the self-force is usually very small compared to other forces acting on the particle. This is because the self-force is proportional to the square of the charge of the particle and inversely proportional to the square of the distance between different parts of the particle. For a typical charged particle, such as an electron, the self-force is overwhelmed by other forces, such as electromagnetic forces from external sources or forces from interactions with other charged particles.
The self-force becomes more significant in extreme situations involving very small distances or very high energies. For example, in the context of quantum field theory and particle physics, when considering interactions at extremely short distances or when particles have extremely high energies, the self-force cannot be ignored. In these cases, a more precise treatment, which takes into account the self-force, is required.
In summary, while a charged particle does experience a self-force due to its own electric field when it's in motion, this force is often negligible compared to other forces acting on the particle and can be safely ignored in many practical situations.