In a cyclotron we increase the K.E. of a particle by allowing the E field to do work on it. But if I take the cyclotron and the particle as my system it's net energy only increases. How will the net energy be conserved here ??!
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A cyclotron consists of two large "D" shaped cavities between large powerful electromagnets. This is the first part where external power is needed, as electromagnets require a current to operate at their needed magnetic strength. The magnetic field is only there to maintain the particles in orbits so they do not exit the cyclotron too early. When the particles first enter, the Dees have an alternating voltage between them to accelerate the charges into one of the cavities, they will now follow an ever widening circular orbit due to the Lorentz force from the magnets and alternating electric field. The electric field that oscillates must do so at the cyclotron resonance frequency $f=\frac{qB}{2\pi m}$. Switching such an electric field at that frequency takes power to do so, and so it takes power to alternate the electric field to aid in accelerating the particles. In this sense work is done by the power grid to the systems that maintain the particle tracks and accelerate them.
