# Proton inside a cyclotron

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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