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$$N=\frac{{q}^{2}{B}^{2}{R}^{2}}{4mqV}$$
and
$${K}_{\text{max}}=2qNV$$
Clearly, the numbers of revolutions is inversely proportional to the potential difference applied in the gap. On the other hand, maximum energy of the particle is directly proportional to the product “NV”. Combining two facts, we find that energy of the particle is indeed independent of the applied voltage in the gap.
We have already noted two limitations of cyclotron as accelerator. One limitation is that it can not accelerate neutral particle. Second limitation is that lighter elementary particles like electrons or positrons can not be accelerated and requires important changes or modifications of the device. In addition to these, there are two other important limitations as described here.
The relativistic effect becomes significant enough to be neglected when particle achieve 10 % of the speed of light. The energy corresponding to this speed for a proton is about 5MeV. Initially, the small relativistic effect is accommodated by an standard cyclotron, but it begins to fail to accelerate charged particle at higher energy level of 50 MeV or so.
At higher speed, the mass of the particle increases in accordance with following equation :
$$m=\frac{{m}_{0}}{\sqrt{\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}}$$
where mo is rest mass and c is the speed of light in vacuum. The particle becomes heavier at higher speed. Putting this in the expression of frequency, we have :
$$\Rightarrow \nu =\frac{qB\sqrt{\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}}{2\pi {m}_{0}}$$ $$\Rightarrow \nu ={\nu}_{0}\sqrt{\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}$$
where ${\nu}_{0}$ is classical frequency. Clearly, the frequency of revolution decreases with increasing velocity whereas frequency of applied electrical oscillator is fixed. The particle, therefore, gets out of step with the alternating electrical field. As a result, speed of the particle does not increase beyond a certain value.
The cyclotron is also limited by the mere requirement of magnet size as radius of Dees increases with increasing speed of the particle being accelerated. Let us calculate speed corresponding of a 100 GeV particle in a magnetic field of 1 T. The radius of revolution is related to kinetic energy :
$${K}_{\text{max}}=\frac{{q}^{2}{B}^{2}{R}^{2}}{2m}$$ $$\Rightarrow R=\sqrt{\left(\frac{2m{K}_{\text{max}}}{{q}^{2}{B}^{2}}\right)}$$
The given kinetic energy is :
$$\Rightarrow {K}_{\text{max}}=100X{10}^{9}\phantom{\rule{1em}{0ex}}eV={10}^{11}X1.6X{10}^{-19}=1.6X{10}^{-8}\phantom{\rule{1em}{0ex}}J$$
Now, putting values assuming particle to be a proton,
$$\Rightarrow R=\sqrt{\left(\frac{2X1.66X{10}^{-27}X1.6X{10}^{-8}}{{\left(1.6X{10}^{-19}\right)}^{2}X1}\right)}$$ $$\Rightarrow R=0.144X{10}^{2}=14.4m$$
We can imagine how costly it would be to create magnet of such an extent. For higher energy, the required radius could be in kilometers.
The synchrocyclotron is a device that addresses the limitation due to relativistic effect. The frequency of oscillator is reduced gradually in order to maintain the resonance with the spiral motion of charged particle. Note that magnetic field remains constant as in the case of cyclotron.
In synchrotron as against synchrocyclotron, both magnetic field and electric field are variable. It aims to address both the limitations due to relativistic effect as well as due to the requirement of large cross section of magnets. The particle is accelerated along a fixed large circular path inside a torus shaped tunnel. The magnetic field here bends the particle, where as electric field changes speed. Clearly, the requirement of a large cross section of magnet is converted into multiple bending magnets along a large radius fixed circular path.
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