A cyclotron adjusted to give proton beam, magnetic induction is `0.15wbm^(-2)` and the extreme radius is 1.5m. The energy of emergent proton in Me V will be

Options

A. 3.42
B. 2.43
C. 24.3
D. 34.2

Answer

To solve the problem of finding the energy of the emergent proton in a cyclotron, we can follow these steps:

### Step 1: Understand the relationship between radius, velocity, charge, and magnetic field
The radius \( r \) of a proton's path in a cyclotron is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the proton,
- \( v \) is the velocity of the proton,
- \( q \) is the charge of the proton,
- \( B \) is the magnetic induction.

### Step 2: Rearranging the formula for velocity
From the equation above, we can express the velocity \( v \) as:
\[ v = \frac{qBr}{m} \]

### Step 3: Kinetic energy of the proton
The kinetic energy \( KE \) of the proton can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the expression for \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left( \frac{qBr}{m} \right)^2 \]
This simplifies to:
\[ KE = \frac{q^2 B^2 r^2}{2m} \]

### Step 4: Substitute the known values
Now we can substitute the known values into the equation:
- Charge of proton \( q = 1.6 \times 10^{-19} \, \text{C} \)
- Magnetic induction \( B = 0.15 \, \text{Wb/m}^2 \)
- Radius \( r = 1.5 \, \text{m} \)
- Mass of proton \( m = 1.67 \times 10^{-27} \, \text{kg} \)

Substituting these values into the kinetic energy formula:
\[ KE = \frac{(1.6 \times 10^{-19})^2 (0.15)^2 (1.5)^2}{2 \times (1.67 \times 10^{-27})} \]

### Step 5: Calculate the kinetic energy
Calculating each part:
1. \( (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \)
2. \( (0.15)^2 = 0.0225 \)
3. \( (1.5)^2 = 2.25 \)
4. \( 2 \times (1.67 \times 10^{-27}) = 3.34 \times 10^{-27} \)

Now substituting these values:
\[ KE = \frac{2.56 \times 10^{-38} \times 0.0225 \times 2.25}{3.34 \times 10^{-27}} \]

Calculating the numerator:
\[ 2.56 \times 10^{-38} \times 0.0225 \times 2.25 = 1.287 \times 10^{-39} \]

Now calculating:
\[ KE = \frac{1.287 \times 10^{-39}}{3.34 \times 10^{-27}} \approx 3.86 \times 10^{-13} \, \text{J} \]

### Step 6: Convert Joules to MeV
To convert Joules to MeV, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ KE \, \text{(in MeV)} = \frac{3.86 \times 10^{-13}}{1.6 \times 10^{-13}} \approx 2.4 \, \text{MeV} \]

### Final Answer
The energy of the emergent proton is approximately \( 2.4 \, \text{MeV} \).

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