Application in cyclotrons
The application of the above in cyclotrons is immediate:[33][34][35][36][37][38][39][40][41][42][43]
In the presence of a magnetic field only, the Lorentz force is:
Since:
it follows that:
meaning that γ is constant, and so is v. This is instrumental in solving the equation of motion for a charge particle of charge q in a magnetic field of induction B as follows:
On the other hand:
Thus:
Separating by components, we obtain:
The solutions are:
By integrating one more time with respect to t the differential equations above we obtain the equations of motion: a circle of radius in the plane z=constant, where v0 is the initial speed of the particle entering the cyclotron. Notice that this calculation ignores the Abraham-Lorentz force In the physics of electromagnetism, the Abraham-Lorentz force is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force which is the reaction to the emission of electromagnetic radiation by the particle. If the speed is held constant by applying an electric field, then the magnitude of the acceleration is constant, but its direction keeps changing in a cyclotron. The jerk is proportional with the second time derivative of speed:
Because the jerk In physics, jerk, also known as jolt , surge and lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, the second derivative of velocity, or the third derivative of position. Jerk is defined by the following equation: is directed opposite to the velocity, the Abraham-Lorentz force tends to slow the particle down. Note that the Abraham-Lorentz force is much smaller than the Lorentz force In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields::
so, it can be ignored in most computations.
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