A proton and an alpha particle enter the same magnetic field which is perpendicular to their velocity. If they have same kinetic energy then ratio of radii of their circular path is

Options

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

Answer

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

### Step 1: Understand the formula for the radius of circular motion in a magnetic field
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 2: Define the parameters for the proton and alpha particle
- For a **proton**:
- Mass \( m_p = m \)
- Charge \( q_p = e \) (where \( e \) is the elementary charge)

- For an **alpha particle**:
- Mass \( m_{\alpha} = 4m \) (since an alpha particle consists of 2 protons and 2 neutrons)
- Charge \( q_{\alpha} = 2e \)

### Step 3: Set up the kinetic energy equation
Since both particles have the same kinetic energy \( KE \):
\[ KE_p = KE_{\alpha} \]
\[ \frac{1}{2} m_p v_p^2 = \frac{1}{2} m_{\alpha} v_{\alpha}^2 \]

### Step 4: Substitute the masses
Substituting the masses:
\[ \frac{1}{2} m v_p^2 = \frac{1}{2} (4m) v_{\alpha}^2 \]
Cancelling \( \frac{1}{2} \) and \( m \) from both sides gives:
\[ v_p^2 = 4 v_{\alpha}^2 \]
Taking the square root:
\[ v_p = 2 v_{\alpha} \]

### Step 5: Calculate the radii for both particles
Now, substituting the values into the radius formula:

- For the proton:
\[ r_p = \frac{mv_p}{q_p B} = \frac{m(2v_{\alpha})}{eB} = \frac{2mv_{\alpha}}{eB} \]

- For the alpha particle:
\[ r_{\alpha} = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha} B} = \frac{(4m)v_{\alpha}}{2eB} = \frac{4mv_{\alpha}}{2eB} = \frac{2mv_{\alpha}}{eB} \]

### Step 6: Calculate the ratio of the radii
Now we can find the ratio of the radii:
\[ \frac{r_p}{r_{\alpha}} = \frac{\frac{2mv_{\alpha}}{eB}}{\frac{2mv_{\alpha}}{eB}} = 1 \]

### Conclusion
Thus, the ratio of the radii of their circular paths is:
\[ \frac{r_p}{r_{\alpha}} = 1:1 \]