Two particles `X` and `Y` with equal charges, after being accelerated throuhg the same potential difference, enter a region of uniform magnetic field and describe circular paths of radii `R_(1)` and `R_(2)` respectively. The ratio of the mass of `X` to that of `Y` is

Options

A. `R_(1) //R_(2)`
B. `(R_(1)//R_(2))^(2)`
C. `(R_(2)//R_(1))`
D. `(R_(2)//R_(2))^(2)`

Explanation

To solve the problem of finding the ratio of the masses of two particles \(X\) and \(Y\) that are accelerated through the same potential difference and move in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion in Magnetic Field:**
When charged particles move in a magnetic field, they experience a magnetic force that acts as a centripetal force, causing them to move in circular paths. The centripetal force is given by:
\[ F_c = \frac{mv^2}{r} \]
where \(m\) is the mass of the particle, \(v\) is its velocity, and \(r\) is the radius of the circular path.

2. **Magnetic Force:**
The magnetic force acting on a charged particle moving in a magnetic field is given by:
\[ F_m = qvB \]
where \(q\) is the charge of the particle, \(v\) is its velocity, and \(B\) is the magnetic field strength.

3. **Equating Forces:**
Since the magnetic force provides the centripetal force, we can set them equal to each other:
\[ \frac{mv^2}{r} = qvB \]

4. **Solving for Velocity:**
Rearranging the equation gives:
\[ mv = qBr \]
Thus, the velocity \(v\) can be expressed as:
\[ v = \frac{qBr}{m} \]

5. **Kinetic Energy Relation:**
The kinetic energy \(K\) of the particle is given by:
\[ K = \frac{1}{2} mv^2 \]
When the particle is accelerated through a potential difference \(V\), the kinetic energy gained is equal to the work done on it:
\[ K = qV \]

6. **Equating Kinetic Energy:**
Setting the two expressions for kinetic energy equal gives:
\[ \frac{1}{2} mv^2 = qV \]

7. **Substituting Velocity:**
Substitute \(v\) from step 4 into the kinetic energy equation:
\[ \frac{1}{2} m \left(\frac{qBr}{m}\right)^2 = qV \]
Simplifying this leads to:
\[ \frac{q^2B^2r^2}{2m} = qV \]

8. **Rearranging for Mass:**
Rearranging gives:
\[ m = \frac{q^2B^2r^2}{2qV} = \frac{qBr^2}{2V} \]

9. **Finding the Mass Ratio:**
For particles \(X\) and \(Y\) with radii \(R_1\) and \(R_2\) respectively, we can write:
\[ \frac{m_X}{m_Y} = \frac{qB R_1^2 / 2V}{qB R_2^2 / 2V} = \frac{R_1^2}{R_2^2} \]

10. **Final Result:**
Thus, the ratio of the masses of particles \(X\) and \(Y\) is:
\[ \frac{m_X}{m_Y} = \frac{R_1^2}{R_2^2} \]

### Conclusion:
The ratio of the mass of particle \(X\) to that of particle \(Y\) is given by:
\[ \frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2 \]