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In the given figure of a cyclotron, showing the particle source \( \mathrm{S} \) 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 energy every time they cross the gap between the dees.
Beame
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 aceelerate 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
W feld does not enter the dee. The magnetic field, however, is not screened by the (nonmagnctic) copper dee, so the proton moves in circular path whose radius, which depends on its speed, is given by
Ratio of radius of successive semi circular path
\[
\text { Eq. } r-\frac{m v}{q B} \quad-(1)
\]
(A) \( \sqrt{1}: \sqrt{2}: \sqrt{3}: \sqrt{4} \)
Let us assume that at the instant the proton emerges
(B) \( \sqrt{1}: \sqrt{3}: \sqrt{5} \)
(C) \( \sqrt{2}: \sqrt{4}: \sqrt{6} \) proton again faces a negatively charged dee and is again aceelerated. Thus, the proton again faces a
(D) \( 1: 2: 3 \) in step, with the eseillations 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 cyclotron is that the frequency \( \mathrm{f} \) at which the proton circulates in the field (and that does not depend on its speed) must be equal to the fixed frequency \( \mathrm{f} \) - of the electrical oscillator, or \( \mathrm{f}=\mathrm{r} \) _- (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 \( f \). - 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 \( \mathrm{qB}=2 \pi \mathrm{mf} \)
For the proton, \( q \) and \( m \) are fixed. The oscillator (we assume) is designed to work at a single fixed frequency C-. We then "tune" the cyclotron by varying B until through the magnetic feld, to emerge as a beam.
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