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In the given figure of a cyclotron, showing the particle source \( \mathbf{S} \) and the dees. A uniform magnetic field is directed up from the plane of the page. Circulating protons spiral outward within the hollow dees, gaining encrgy every time they cross the gap between the dees.
Suppose that a proton, injected by source \( S \) at the centre of the cyclotron in Fig., initially moves toward a negatively charged dee. It will accelerate toward this dee and enter it. Once inside, it is shielded from electric field by the copper walls of the dee, that is the electric field does not enter the dee. The magnetic field, however, is not sereened by the (nonmagnetic) copper dee, so the proton moves in circular path whose radius, Change in kinetic energy of charge particle after every which depends on its speed, is given by
PV time period is :
(A) \( 2 q \mathrm{q} \)
(B) \( \mathrm{qV} \) into the center ap from the firs dee, the potential
(C) \( 3 q \mathrm{~V} \)
(D) None of these again aceelerated. Thus, the proton again faces a negatively charged dee and is azain aceelerated. This process continues, the circulating proton always being in step, with the oseillations of the dee potential, until the proton has spiraled out to the edge of the dee system. There a deflector plate sends it out through a portal.

The key to the operation of the eyclotron is that the frequency \( f \) at which the proton eirculates in the feld (and that does not depend on its speed) must be equal to the fixed frequency \( \mathrm{f} \) - of the electrical oscillator, or \( \mathrm{f}=\mathrm{f} \) _-(resonance condition).
(2)
This resonance condition says that, if the energy of the circulating proton is to increase, energy must be fed to it at a frequency \( r \) - that is equal to the natural frequency \( r \) at which the proton circulates in the magnetic field.
Combining Eq. 1 and 2 allows us to write the rewonance condition as
\( \mathrm{qB}=2 \pi \mathrm{mf} \)
(3)
For the proton, \( q \) and \( m \) are fixed. The oscillator (we assume) is designed to work at a single fixed fiequency C.-. We then "tune" the eyclotron by varying B until eq. 3 is satisfied and then many protons circulate through the magnetic field, to emerge as a beam.
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