An electron, a proton and a deuteron move in a magnetic field with same momentum perpendicularly. The ratio of the radii of their circular paths will be -

Options

A. `1 : 1 : 1`
B. `1 : 1 : 2`
C. `1 : 2 : 4`
D. `2 : 1 : 1`

Answer

To find the ratio of the radii of the circular paths of an electron, a proton, and a deuteron moving in a magnetic field with the same momentum perpendicularly, we can follow these steps:

### Step 1: Understand the motion of charged particles in a magnetic field
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts perpendicular to both its velocity and the magnetic field. This results in circular motion.

### Step 2: Write the expression for the radius of the circular path
The radius \( r \) of the circular path of a charged particle 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 3: Substitute momentum into the formula
The momentum \( p \) of a particle is defined as:
\[ p = mv \]
Thus, we can express the radius in terms of momentum:
\[ r = \frac{p}{qB} \]

### Step 4: Analyze the given particles
- **Electron**: Charge \( q_e = -e \), mass \( m_e \)
- **Proton**: Charge \( q_p = +e \), mass \( m_p \)
- **Deuteron**: Charge \( q_d = +e \), mass \( m_d \) (approximately twice the mass of a proton)

### Step 5: Calculate the ratio of the radii
Since the momentum \( p \) and magnetic field \( B \) are the same for all three particles, we can write the ratio of their radii as:
\[ \frac{r_e}{r_p} = \frac{p/q_e B}{p/q_p B} = \frac{q_p}{q_e} \]
\[ \frac{r_d}{r_p} = \frac{p/q_d B}{p/q_p B} = \frac{q_p}{q_d} \]

Since the magnitudes of the charges \( |q_e| = |q_p| = |q_d| = e \), we have:
\[ \frac{r_e}{r_p} = 1 \quad \text{and} \quad \frac{r_d}{r_p} = 1 \]

### Step 6: Final ratio of the radii
Thus, the ratio of the radii of the circular paths of the electron, proton, and deuteron is:
\[ r_e : r_p : r_d = 1 : 1 : 1 \]

### Conclusion
The final answer is that the ratio of the radii of their circular paths will be \( 1 : 1 : 1 \).

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