A beam of 30 MeV `alpha` particles is to be obtained from a cyclotron of radius 50cm . The strength of magnetic field required to be applied will be

Options

A. 1.582 T
B. 0.01582 T
C. 0.1582 T
D. 15.82 T

Answer

To find the strength of the magnetic field required to accelerate a beam of 30 MeV alpha particles in a cyclotron of radius 50 cm, we can follow these steps:

### Step 1: Understand the relationship in a cyclotron
The radius \( r \) of the cyclotron is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) = mass of the particle
- \( v \) = velocity of the particle
- \( q \) = charge of the particle
- \( B \) = magnetic field strength

### Step 2: Calculate the kinetic energy in joules
The kinetic energy \( K.E. \) of the alpha particles is given as 30 MeV. We need to convert this into joules:
\[ K.E. = 30 \, \text{MeV} = 30 \times 10^6 \, \text{eV} = 30 \times 10^6 \times 1.6 \times 10^{-19} \, \text{J} = 4.8 \times 10^{-12} \, \text{J} \]

### Step 3: Find the velocity of the alpha particles
We can use the kinetic energy formula:
\[ K.E. = \frac{1}{2} mv^2 \]
From this, we can solve for \( v \):
\[ v = \sqrt{\frac{2 \times K.E.}{m}} \]

### Step 4: Determine the mass of the alpha particle
The mass of an alpha particle is approximately 4 times the mass of a proton:
\[ m = 4 \times (1.67 \times 10^{-27} \, \text{kg}) = 6.68 \times 10^{-27} \, \text{kg} \]

### Step 5: Substitute values to find velocity
Substituting the values into the velocity equation:
\[ v = \sqrt{\frac{2 \times 4.8 \times 10^{-12}}{6.68 \times 10^{-27}}} \]

### Step 6: Calculate the velocity
Calculating the above expression gives:
\[ v \approx \sqrt{\frac{9.6 \times 10^{-12}}{6.68 \times 10^{-27}}} \approx \sqrt{1.44 \times 10^{15}} \approx 1.2 \times 10^7 \, \text{m/s} \]

### Step 7: Determine the charge of the alpha particle
The charge of an alpha particle is twice the charge of a proton:
\[ q = 2 \times (1.6 \times 10^{-19} \, \text{C}) = 3.2 \times 10^{-19} \, \text{C} \]

### Step 8: Rearrange the cyclotron formula to find B
Rearranging the formula for \( B \):
\[ B = \frac{mv}{qr} \]

### Step 9: Substitute all known values
Substituting the known values into the equation:
- \( m = 6.68 \times 10^{-27} \, \text{kg} \)
- \( v = 1.2 \times 10^7 \, \text{m/s} \)
- \( q = 3.2 \times 10^{-19} \, \text{C} \)
- \( r = 0.5 \, \text{m} \)

\[ B = \frac{(6.68 \times 10^{-27})(1.2 \times 10^7)}{(3.2 \times 10^{-19})(0.5)} \]

### Step 10: Calculate B
Calculating the above expression gives:
\[ B \approx \frac{8.016 \times 10^{-20}}{1.6 \times 10^{-19}} \approx 0.501 \, \text{T} \approx 1.5 \, \text{T} \]

### Final Answer
The strength of the magnetic field required is approximately:
\[ B \approx 1.5 \, \text{T} \]