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Question: A proton is accelerating in a cyclotron where the applied magnetic field is \(2\;{\text{T}}\). If th...

A proton is accelerating in a cyclotron where the applied magnetic field is 2  T2\;{\text{T}}. If the potential gap effectively 100kV100{\text{kV}} then how much revolutions the proton has to make between the "dees" to acquire a kinetic energy of 20MeV?20{\text{MeV}}?
(A) 100100
(B) 150150
(C) 200200
(D) 300300

Explanation

Solution

In this question, we will proceed by finding all the values in SI units included in the given data. In order to directly get the required result, we will use a simple formula.
Formula Used: We will use the following formula for the kinetic energy acquired by a charge when it is revolving between the dees
KE=2NqV{\text{KE}} = 2NqV

Complete Step-by-Step Solution:
The following information is provided to us in the question
The applied magnetic field, B=2TB = 2T
The potential difference, V=100KV=100×1000  V=105  VV = 100{\text{KV}} = 100 \times 1000\;{\text{V}} = {10^5}\;{\text{V}}
The kinetic energy to be acquired is given as KE=20MeV=20×106eV{\text{KE}} = 20{\text{MeV}} = 20 \times {10^6}{\text{eV}}
Where
e=1.6×10−19Ce = 1.6 \times {10^{ - 19}}{\text{C}}(charge of a proton)
So, we get
KE=20×106×1.6×10−19  V{\text{KE}} = 20 \times {10^6} \times 1.6 \times {10^{ - 19}}\;{\text{V}}
Now, we will use the formula listed above
KE=2NqV{\text{KE}} = 2NqV
By substituting the values of the kinetic energy, charge and the potential difference in the above equation, we get
⇒20×106×1.6×10−19=2N(1.6×10−19)105\Rightarrow 20 \times {10^6} \times 1.6 \times {10^{ - 19}} = 2N\left( {1.6 \times {{10}^{ - 19}}} \right){10^5}
⇒N=20×106×(1.6×10−19)2(1.6×10−19)105\Rightarrow N = \dfrac{{20 \times {{10}^6} \times \left( {1.6 \times {{10}^{ - 19}}} \right)}}{{2\left( {1.6 \times {{10}^{ - 19}}} \right){{10}^5}}}
Upon further solving, we get
∴N=100\therefore N = 100

So, 100100 revolutions are required to acquire a kinetic energy of 20MeV20 MeV
Hence, the correct option is (A.)

Additional Information: A cyclotron is a type of accelerator for compact particles that produces radioactive isotopes that can be used for imaging procedures. In a magnetic field, stable, non-radioactive isotopes are placed into the cyclotron, which accelerates charged particles (protons) to high energy.

Note: In this specific problem, an additional data is given, i.e., the magnitude of the magnetic field equal to 2T2 T. To solve this issue, this information is not at all necessary. The units should be taken care of in these types of challenges. In order to ensure the units' transparency, we converted kilovolts into volts and MeV into volts.