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

Answer

To solve the problem, we need to find the ratio of the masses of two particles \(X\) and \(Y\) that have equal charges and are accelerated through the same potential difference before entering a uniform magnetic field. The particles describe circular paths with radii \(R_1\) and \(R_2\) respectively.

### Step-by-Step Solution:

1. **Understanding the Motion in Magnetic Field**:
The radius \(R\) of the circular path of a charged particle moving in a magnetic field 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

2. **Kinetic Energy and Velocity**:
The kinetic energy \(K\) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
When the particle is accelerated through a potential difference \(V\), the kinetic energy can also be expressed as:
\[ K = qV \]
Setting these two expressions for kinetic energy equal gives:
\[ \frac{1}{2} mv^2 = qV \]
From this, we can solve for the velocity \(v\):
\[ v = \sqrt{\frac{2qV}{m}} \]

3. **Substituting Velocity into the Radius Formula**:
Now we substitute the expression for \(v\) back into the radius formula:
\[ R = \frac{m \cdot \sqrt{\frac{2qV}{m}}}{qB} \]
Simplifying this, we get:
\[ R = \frac{\sqrt{2m qV}}{qB} \]
This shows that the radius \(R\) is proportional to \(\sqrt{m}\):
\[ R \propto \sqrt{m} \]

4. **Setting Up Ratios for Particles \(X\) and \(Y\)**:
For particles \(X\) and \(Y\), we have:
\[ R_1 \propto \sqrt{m_X} \quad \text{and} \quad R_2 \propto \sqrt{m_Y} \]
Therefore, we can write:
\[ \frac{R_1}{R_2} = \frac{\sqrt{m_X}}{\sqrt{m_Y}} \]

5. **Squaring Both Sides**:
To find the ratio of the masses, we square both sides:
\[ \left(\frac{R_1}{R_2}\right)^2 = \frac{m_X}{m_Y} \]
Hence, the ratio of the masses of particles \(X\) and \(Y\) is:
\[ \frac{m_X}{m_Y} = \frac{R_1^2}{R_2^2} \]

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