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. MeV?

Options

A. `50`
B. `100`
C. `150`
D. `200`

Explanation

To solve the problem of how many revolutions protons must complete in a cyclotron to acquire a kinetic energy of 20 MeV, we can follow these steps:

### Step 1: Understand the relationship between potential difference and kinetic energy
In a cyclotron, when a charged particle (like a proton) is accelerated through a potential difference (P.D.), it gains kinetic energy (K.E.). The kinetic energy gained by a charged particle can be calculated using the formula:
\[ K.E. = q \cdot V \]
where \( q \) is the charge of the particle and \( V \) is the potential difference.

### Step 2: Calculate the charge of a proton
The charge of a proton (\( q \)) is approximately:
\[ q = 1.6 \times 10^{-19} \text{ C} \]

### Step 3: Convert the potential difference to joules
Given that the potential difference across the dees is 100 kV, we convert this to volts:
\[ V = 100 \times 10^3 \text{ V} = 100,000 \text{ V} \]

### Step 4: Calculate the kinetic energy gained after one revolution
Now, we can calculate the kinetic energy gained after one complete revolution:
\[ K.E. = q \cdot V = (1.6 \times 10^{-19} \text{ C}) \cdot (100,000 \text{ V}) = 1.6 \times 10^{-14} \text{ J} \]

### Step 5: Convert kinetic energy from joules to MeV
To convert joules to MeV, we use the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \):
\[ K.E. = \frac{1.6 \times 10^{-14} \text{ J}}{1.6 \times 10^{-19} \text{ J/eV}} = 10^5 \text{ eV} = 0.1 \text{ MeV} \]

### Step 6: Determine the number of revolutions needed for 20 MeV
Now, we need to find out how many revolutions are required to achieve a kinetic energy of 20 MeV:
\[ \text{Number of revolutions} = \frac{\text{Total K.E. desired}}{\text{K.E. gained per revolution}} = \frac{20 \text{ MeV}}{0.1 \text{ MeV}} = 200 \]

### Conclusion
Thus, the protons must complete **200 revolutions** to acquire a kinetic energy of 20 MeV.