An `alpha` particle and a proton having same momentum enter into a region of uniform magnetic field and move in circular paths. The ratio of the radii of curvature of their paths, `(R_(alpha))/(R_(p))` in the field is

Options

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

Answer

To find the ratio of the radii of curvature of the paths of an alpha particle and a proton in a uniform magnetic field, we start with the formula for the radius of curvature \( R \) of a charged particle moving in a magnetic field:

\[ R = \frac{mV}{QB} \]

where:
- \( R \) is the radius of curvature,
- \( m \) 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 1: Write the expression for the radius of curvature for the proton.

For a proton:
- Mass \( m_p = m \) (let's denote the mass of the proton as \( m \)),
- Charge \( Q_p = Q \) (the charge of the proton is \( +e \)).

Thus, the radius of curvature for the proton \( R_p \) is given by:

\[ R_p = \frac{mV}{QB} \]

### Step 2: Write the expression for the radius of curvature for the alpha particle.

For an alpha particle:
- Mass \( m_{\alpha} = 4m \) (the mass of an alpha particle is approximately four times that of a proton),
- Charge \( Q_{\alpha} = 2Q \) (the charge of an alpha particle is \( +2e \)).

Thus, the radius of curvature for the alpha particle \( R_{\alpha} \) is given by:

\[ R_{\alpha} = \frac{(4m)V}{(2Q)B} \]

### Step 3: Simplify the expression for the radius of curvature of the alpha particle.

We can simplify \( R_{\alpha} \):

\[ R_{\alpha} = \frac{4mV}{2QB} = \frac{2mV}{QB} \]

### Step 4: Find the ratio of the radii of curvature.

Now, we can find the ratio of the radii of curvature \( \frac{R_{\alpha}}{R_p} \):

\[ \frac{R_{\alpha}}{R_p} = \frac{\frac{2mV}{QB}}{\frac{mV}{QB}} = \frac{2mV}{QB} \cdot \frac{QB}{mV} \]

### Step 5: Cancel out the common terms.

When we cancel out the common terms \( mV \) and \( QB \):

\[ \frac{R_{\alpha}}{R_p} = \frac{2}{1} = 2 \]

### Conclusion

Thus, the ratio of the radii of curvature of the paths of the alpha particle and the proton is:

\[ \frac{R_{\alpha}}{R_p} = 2 \]