A proton , a deuteron and an `alpha`-particle enter a magnetic field perpendicular to field with same velocity. What is the ratio of the radii of circular paths ?

Options

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

Explanation

To find the ratio of the radii of the circular paths of a proton, a deuteron, and an alpha particle when they enter a magnetic field perpendicularly with the same velocity, we can use the following 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 as a centripetal force, causing it to move in a circular path. The magnetic force \( F \) on a charged particle is given by:
\[ F = qvB \]
where:
- \( q \) is the charge of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field strength.

### Step 2: Relate the magnetic force to centripetal force
The magnetic force acts as the centripetal force required for circular motion. The centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the particle,
- \( r \) is the radius of the circular path.

### Step 3: Set the magnetic force equal to the centripetal force
Setting the two forces equal gives us:
\[ qvB = \frac{mv^2}{r} \]
From this, we can solve for the radius \( r \):
\[ r = \frac{mv}{qB} \]

### Step 4: Determine the charge and mass for each particle
- For a **proton**:
- Charge \( q_p = e \) (where \( e \) is the elementary charge),
- Mass \( m_p = m_p \) (mass of the proton).

- For a **deuteron** (which is a nucleus of deuterium, consisting of one proton and one neutron):
- Charge \( q_d = e \),
- Mass \( m_d = 2m_p \) (approximately twice the mass of a proton).

- For an **alpha particle** (which consists of 2 protons and 2 neutrons):
- Charge \( q_{\alpha} = 2e \),
- Mass \( m_{\alpha} = 4m_p \) (approximately four times the mass of a proton).

### Step 5: Write the radius for each particle
Using the formula \( r = \frac{mv}{qB} \):

1. For the proton:
\[ r_p = \frac{m_p v}{eB} \]

2. For the deuteron:
\[ r_d = \frac{(2m_p)v}{eB} = \frac{2m_p v}{eB} \]

3. For the alpha particle:
\[ r_{\alpha} = \frac{(4m_p)v}{2eB} = \frac{4m_p v}{2eB} = \frac{2m_p v}{eB} \]

### Step 6: Calculate the ratio of the radii
Now, we can find the ratios of the radii:
\[ \text{Ratio} = \frac{r_p}{r_d} : \frac{r_d}{r_{\alpha}} : \frac{r_{\alpha}}{r_p} = \frac{m_p v / eB}{2m_p v / eB} : \frac{2m_p v / eB}{2m_p v / eB} : \frac{2m_p v / eB}{m_p v / eB} \]
This simplifies to:
\[ 1 : 2 : 1 \]

### Final Ratio
Thus, the ratio of the radii of the circular paths of a proton, a deuteron, and an alpha particle is:
\[ 1 : 2 : 1 \]