A proton and an alpha particle are separately projected in a region where a uniform magnetic field exists. Their initial velocities are perpendicular to direction of magnetic field. If both the particles move around magnetic field in circles of equal radii, the ratio of momentum of proton to alpha particle `(p_(p)/(p_(alpha)))` is

Options

A. `1`
B. `(1)/(2)`
C. `2`
D. `(1)/(4)`

Explanation

To solve the problem, we need to analyze the motion of both the proton and the alpha particle in a magnetic field. We will derive the necessary equations step by step.

### 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 causes it to move in a circular path. The radius of this circular path (r) 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: Write the equations for the radius of the proton and alpha particle
Let’s denote:
- \( m_p \) = mass of the proton
- \( v_p \) = velocity of the proton
- \( q_p \) = charge of the proton
- \( m_{\alpha} \) = mass of the alpha particle
- \( v_{\alpha} \) = velocity of the alpha particle
- \( q_{\alpha} \) = charge of the alpha particle

According to the given information, both particles move in circles of equal radii. Therefore, we can write:

For the proton:
\[ r = \frac{m_p v_p}{q_p B} \]

For the alpha particle:
\[ r = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha} B} \]

Since the radii are equal, we can set these two equations equal to each other:

\[ \frac{m_p v_p}{q_p B} = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha} B} \]

### Step 3: Simplify the equation
We can cancel \( B \) from both sides:

\[ \frac{m_p v_p}{q_p} = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha}} \]

### Step 4: Express momentum
The momentum \( p \) of a particle is given by:

\[ p = mv \]

Thus, we can express the momentum of the proton and the alpha particle as:

- Momentum of the proton: \( p_p = m_p v_p \)
- Momentum of the alpha particle: \( p_{\alpha} = m_{\alpha} v_{\alpha} \)

### Step 5: Find the ratio of momenta
Now, we need to find the ratio of the momentum of the proton to the momentum of the alpha particle:

\[ \frac{p_p}{p_{\alpha}} = \frac{m_p v_p}{m_{\alpha} v_{\alpha}} \]

From the earlier equation \( \frac{m_p v_p}{q_p} = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha}} \), we can rearrange it to express \( \frac{m_p v_p}{m_{\alpha} v_{\alpha}} \):

\[ \frac{m_p v_p}{m_{\alpha} v_{\alpha}} = \frac{q_p}{q_{\alpha}} \]

### Step 6: Substitute the charges
The charge of a proton \( q_p \) is \( e \) (elementary charge), and the charge of an alpha particle \( q_{\alpha} \) is \( 2e \) (since it consists of 2 protons and 2 neutrons). Therefore, we have:

\[ \frac{q_p}{q_{\alpha}} = \frac{e}{2e} = \frac{1}{2} \]

### Final Result
Thus, the ratio of the momentum of the proton to the momentum of the alpha particle is:

\[ \frac{p_p}{p_{\alpha}} = \frac{1}{2} \]