When a charged particle is projected at an extremely high speed...

When a charged particle moves in a uniform magnetic field perpendicular to its velocity, it experiences a magnetic force that provides the necessary centripetal acceleration for circular motion.

At extremely high speeds, the accelerating charged particle continuously radiates electromagnetic energy in the form of synchrotron radiation.

Due to this continuous loss of energy, the kinetic energy and the momentum $p$ of the particle decrease over time.

The radius of curvature of the path is given by $r = \dfrac{p}{qB}$.

Since the momentum $p$ decreases while the charge $q$ and magnetic field $B$ remain constant, the radius of curvature $r$ also decreases.

Therefore, the particle travels along a curve with a decreasing radius of curvature.