Proton , deuton and alpha particle of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton , deuteron and alpha particle are respectively `r_(p), r_(d) and r_(alpha)`. Which one of the following relation is correct?

Options

A. `r_(a) = r_(p) = r_(d)`
B. `r_(a) = r_(p) lt r_(d)`
C. `r_(a) gt r_(d) gt r_(p)`
D. `r_(a) = r_(d) gt r_(p)`

Answer

To find the relationship between the radii of the circular trajectories of a proton, deuteron, and alpha particle moving in a constant magnetic field with the same kinetic energy, we can follow these steps:

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

### Step 2: Express velocity in terms of kinetic energy
The kinetic energy \( K \) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
From this, we can express the velocity \( v \) as:
\[ v = \sqrt{\frac{2K}{m}} \]

### Step 3: Substitute velocity into the radius formula
Substituting the expression for \( v \) into the radius formula gives:
\[ r = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]

### Step 4: Analyze the particles
For the three particles:
1. **Proton**:
- Mass \( m_p = 1 \, \text{u} \) (atomic mass unit)
- Charge \( q_p = 1e \)

\[ r_p = \frac{\sqrt{2K \cdot 1}}{1eB} = \frac{\sqrt{2K}}{eB} \]

2. **Deuteron**:
- Mass \( m_d = 2 \, \text{u} \)
- Charge \( q_d = 1e \)

\[ r_d = \frac{\sqrt{2K \cdot 2}}{1eB} = \frac{\sqrt{4K}}{eB} = \frac{2\sqrt{K}}{eB} \]

3. **Alpha Particle**:
- Mass \( m_{\alpha} = 4 \, \text{u} \)
- Charge \( q_{\alpha} = 2e \)

\[ r_{\alpha} = \frac{\sqrt{2K \cdot 4}}{2eB} = \frac{\sqrt{8K}}{2eB} = \frac{2\sqrt{2K}}{2eB} = \frac{\sqrt{2K}}{eB} \]

### Step 5: Compare the radii
Now we can summarize the radii:
- \( r_p = \frac{\sqrt{2K}}{eB} \)
- \( r_d = \frac{2\sqrt{K}}{eB} \)
- \( r_{\alpha} = \frac{\sqrt{2K}}{eB} \)

From this, we can see:
- \( r_p = r_{\alpha} \)
- \( r_d > r_p \) and \( r_d > r_{\alpha} \)

### Conclusion
Thus, the correct relation is:
\[ r_p = r_{\alpha} < r_d \]