A proton, a deuteron and an `alpha`-particle with the same KE enter a region of uniform magnetic field, moving at right angles to B. What is the ratio of the radii of their circular paths ?

Options

A. `1 : sqrt(2): 1`
B. `1 : sqrt(2) : sqrt(2)`
C. `sqrt(2) : 1 : 1`
D. `sqrt(2) : sqrt(2) : 1`

Explanation

To solve the problem of finding the ratio of the radii of the circular paths of a proton, a deuteron, and an alpha particle moving in a magnetic field with the same kinetic energy, we can follow these steps:

### Step 1: Understand the motion in a magnetic field
When a charged particle moves in a magnetic field at right angles to the field, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. The magnetic force \( F \) is given by:
\[ F = QVB \]
where \( Q \) is the charge of the particle, \( V \) is its velocity, and \( B \) is the magnetic field strength.

### Step 2: Relate the magnetic force to centripetal force
The centripetal force required for circular motion is given by:
\[ F = \frac{MV^2}{R} \]
where \( M \) is the mass of the particle, \( V \) is its velocity, and \( R \) is the radius of the circular path.

### Step 3: Equate the forces
Setting the magnetic force equal to the centripetal force, we have:
\[ QVB = \frac{MV^2}{R} \]

### Step 4: Solve for the radius \( R \)
Rearranging the equation to solve for \( R \), we get:
\[ R = \frac{MV}{QB} \]

### Step 5: Express \( MV \) in terms of kinetic energy
The kinetic energy \( KE \) of the particle is given by:
\[ KE = \frac{1}{2} MV^2 \]
From this, we can express \( MV \) as:
\[ MV = \sqrt{2 \times KE \times M} \]

### Step 6: Substitute \( MV \) into the radius equation
Substituting \( MV \) into the radius equation, we have:
\[ R = \frac{\sqrt{2 \times KE \times M}}{QB} \]

### Step 7: Analyze the three particles
We have three particles:
1. **Proton**: Mass \( M_p \), Charge \( Q_p = e \)
2. **Deuteron**: Mass \( M_d = 2M_p \), Charge \( Q_d = e \)
3. **Alpha particle**: Mass \( M_{\alpha} = 4M_p \), Charge \( Q_{\alpha} = 2e \)

### Step 8: Write the radius for each particle
Using the expression for \( R \):
- For the proton:
\[ R_p = \frac{\sqrt{2 \times KE \times M_p}}{eB} \]
- For the deuteron:
\[ R_d = \frac{\sqrt{2 \times KE \times 2M_p}}{eB} = \frac{\sqrt{4 \times KE \times M_p}}{eB} = 2 \cdot \frac{\sqrt{2 \times KE \times M_p}}{eB} = 2R_p \]
- For the alpha particle:
\[ R_{\alpha} = \frac{\sqrt{2 \times KE \times 4M_p}}{2eB} = \frac{2\sqrt{2 \times KE \times M_p}}{2eB} = \frac{\sqrt{2 \times KE \times M_p}}{eB} = R_p \]

### Step 9: Find the ratio of the radii
Now we can find the ratio of the radii:
\[ R_p : R_d : R_{\alpha} = R_p : 2R_p : R_p = 1 : 2 : 1 \]

### Final Answer
The ratio of the radii of their circular paths is:
\[ 1 : 2 : 1 \]