Proton inside a cyclotron

Electricity and Magnetism Level 3

Consider a standard cyclotron accelerator in which two semi-circular regions (dees) are connected to an AC voltage source which provides an electric field \(E\) and a uniform vertical (out of the page) magnetic \(B\) field inside cyclotron which is perpendicular to the electric field. The electric field switches such that is always points toward the dee the particle is not in.

A proton is released from rest such that it starts rotating in the cyclotron at radius \(R\) and finally comes out from the slits of the cyclotron. The distance between the two semi-circular regions is \(d\). Find the maximum number of turns proton take before coming out from slit's.

Details and Assumptions

\({ { m }_{ p }=1.6\times { 10 }^{ -27 }\text{ kg}\\ { q }_{ p }=1.6\times { 10 }^{ -19 } C\\ B={ 10 }^{ -4 }\text{ T}\\ R=6\text{ m}\\ E=10\frac{\text{V}}{\text{m}}\\ d=10\text{ cm}\\ }\).


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