Here are some of the Limitations of Cyclotron listed below:
(1) When a charged particle is accelerated, its mass also starts increasing with increase in its speed. When its speed become comparable to that of light, the mass of the charged particle become quite large as compared to its rest mass.
If $\mathrm{m}_{0}$ = rest mass,
mass of charged particle moving with speed $v$ is $\quad m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$
Substituting for $\mathrm{m}$ in equation, we have
time spent inside a dee, $t=\frac{\pi \mathrm{m}_{0}}{\mathrm{qB} \sqrt{1-\frac{\mathrm{v}^{2}}{\mathrm{c}^{2}}}}$
Therefore, as v increases, t also increases i.e. the charged particle starts taking more and more time to complete the semi-circular path inside the dee.
Since electric field changes the polarity of the dees after a fixed interval, the charged particle starts lagging behind the electric field and it us ultimately lost by colliding against the walls of the dees.
However, this problem is overcome in the following two ways :
(a) As $v$ increases, $\sqrt{1-\frac{v^{2}}{c^{2}}}$ decreases. Therefore, $B$ is increased in such a manner that the facror $\mathrm{B} \sqrt{1-\frac{\mathrm{v}^{2}}{\mathrm{c}^{2}}}$ and hence t always remains constatn. Such a cyclotron, in which the strength of magnetic field is adjusted to eovercome the problem due to relative variation in mass of the positive ion, is called synchrotron.
(b) The frequency of revolution of charged [particle inside the dees] may be expressed as
$v=B q \sqrt{\frac{1-\frac{v^{2}}{c^{2}}}{2 \pi m_{0}}}$
It follows that as $v$ increases, $\sqrt{1-\frac{v^{2}}{c^{2}}}$ decreases and hence v decreases. If frequency of the electric field is adjusted to be always equal to the frequency of revolution of the charged particle, then such a cyclotron is called synchro-cyclotron or frequency modulated cyclotron.
(2) Cyclotron is used to accelerate heavy charged particles, such as protons. It is not suitable for accelerating electrons.
The reason is that due to small mass, the electrons gain in speed quickly and likewise the relativistic variation in mass quickly makes them out of step with the oscillating electric field.
Also Read:
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