Magnetic field applied on a cyclotron is `0*7T` and radius of its dees is `1*8m`. What will be the energy of the emergent protons in `MeV`? Mass of proton`=1*67xx10^-27kg`.

Explanation

To find the energy of the emergent protons from a cyclotron, we can use the formula for the energy of a charged particle in a magnetic field:

\[ E = \frac{q^2 B^2 r^2}{2m} \]

Where:
- \(E\) is the energy of the proton,
- \(q\) is the charge of the proton,
- \(B\) is the magnetic field strength,
- \(r\) is the radius of the cyclotron's dees,
- \(m\) is the mass of the proton.

### Step 1: Identify the given values
- Magnetic field \(B = 0.7 \, T\)
- Radius \(r = 1.8 \, m\)
- Mass of proton \(m = 1.67 \times 10^{-27} \, kg\)
- Charge of proton \(q = 1.602 \times 10^{-19} \, C\)

### Step 2: Substitute the values into the energy formula
Substituting the known values into the formula:

\[ E = \frac{(1.602 \times 10^{-19})^2 \times (0.7)^2 \times (1.8)^2}{2 \times (1.67 \times 10^{-27})} \]

### Step 3: Calculate the numerator
Calculating each part of the numerator:

1. \(q^2 = (1.602 \times 10^{-19})^2 = 2.5664 \times 10^{-38} \, C^2\)
2. \(B^2 = (0.7)^2 = 0.49 \, T^2\)
3. \(r^2 = (1.8)^2 = 3.24 \, m^2\)

Now, multiply these together:

\[ \text{Numerator} = 2.5664 \times 10^{-38} \times 0.49 \times 3.24 \]

Calculating this gives:

\[ \text{Numerator} = 2.5664 \times 10^{-38} \times 1.588 = 4.079 \times 10^{-38} \, (J \cdot m^2) \]

### Step 4: Calculate the denominator
Calculating the denominator:

\[ \text{Denominator} = 2 \times (1.67 \times 10^{-27}) = 3.34 \times 10^{-27} \]

### Step 5: Calculate the energy \(E\)
Now, divide the numerator by the denominator:

\[ E = \frac{4.079 \times 10^{-38}}{3.34 \times 10^{-27}} = 1.22 \times 10^{-11} \, J \]

### Step 6: Convert energy from Joules to MeV
To convert energy from Joules to Mega electron volts (MeV), use the conversion factor \(1 \, J = 6.242 \times 10^{12} \, MeV\):

\[ E_{MeV} = 1.22 \times 10^{-11} \, J \times 6.242 \times 10^{12} \, MeV/J \]

Calculating this gives:

\[ E_{MeV} \approx 76.2 \, MeV \]

### Final Answer
The energy of the emergent protons is approximately \(76 \, MeV\).

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