Protons are accelerated in a cyclotron where the appilied magnetic field is 2T and the P.D across the dees is 100 KV. How many revolutions the protons has to complete to acquire a K.E. of 20 MeV?

Options

A. 200
B. 300
C. 150
D. 100

Answer

To solve the problem, we need to determine how many revolutions protons must complete in a cyclotron to acquire a kinetic energy of 20 MeV, given a magnetic field of 2 T and a potential difference of 100 kV.

### Step-by-Step Solution:

1. **Understand the Energy Gained per Revolution**:
In a cyclotron, protons gain energy each time they cross the gap between the dees. The potential difference (P.D.) across the dees is given as 100 kV. When protons cross the gap, they gain energy equal to the charge of the proton multiplied by the potential difference.

The energy gained in one crossing (one half-revolution) is:
\[ E_{\text{gain, half}} = q \cdot V \]
where \( q \) is the charge of a proton (approximately \( 1.6 \times 10^{-19} \) C) and \( V = 100 \times 10^3 \) V.

Thus, the energy gained in one full revolution (crossing the gap twice) is:
\[ E_{\text{gain, full}} = 2 \cdot q \cdot V = 2 \cdot (1.6 \times 10^{-19} \, \text{C}) \cdot (100 \times 10^3 \, \text{V}) = 320 \times 10^{-16} \, \text{J} \]

To convert this energy into MeV:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \implies 1 \, \text{MeV} = 1.6 \times 10^{-13} \, \text{J} \]
Therefore,
\[ E_{\text{gain, full}} = \frac{320 \times 10^{-16}}{1.6 \times 10^{-13}} \, \text{MeV} = 2 \, \text{MeV} \]

2. **Total Kinetic Energy Required**:
We need to find out how many revolutions are required to achieve a total kinetic energy of 20 MeV.

3. **Calculate the Number of Revolutions**:
Let \( n \) be the number of revolutions. The total kinetic energy acquired after \( n \) revolutions is:
\[ K.E. = n \cdot E_{\text{gain, full}} = n \cdot 2 \, \text{MeV} \]
Setting this equal to the desired kinetic energy:
\[ n \cdot 2 \, \text{MeV} = 20 \, \text{MeV} \]
Solving for \( n \):
\[ n = \frac{20 \, \text{MeV}}{2 \, \text{MeV}} = 10 \]

### Final Answer:
The number of revolutions the protons must complete to acquire a kinetic energy of 20 MeV is **10 revolutions**.