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 charged particles \(X\) and \(Y\) based on their circular paths in a magnetic field after being accelerated through the same potential difference.

### Step-by-Step Solution:

1. **Understanding 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:
- \(m\) 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.

2. **Relating Velocity to Potential Difference**:
The particles \(X\) and \(Y\) are accelerated through the same potential difference \(V\). The kinetic energy gained by a charged particle when accelerated through a potential difference is given by:
\[ KE = qV \]
This kinetic energy is also expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Equating the two expressions for kinetic energy gives:
\[ qV = \frac{1}{2} mv^2 \implies v = \sqrt{\frac{2qV}{m}} \]

3. **Substituting Velocity into the Radius Formula**:
Substitute the expression for \(v\) into the radius formula:
\[ r = \frac{m \sqrt{\frac{2qV}{m}}}{qB} = \frac{\sqrt{2qVm}}{qB} \]
Thus, we can express the radius in terms of mass:
\[ r = \frac{\sqrt{2qV}}{qB} \cdot \sqrt{m} \]

4. **Setting Up the Ratios for Particles \(X\) and \(Y\)**:
For particle \(X\), the radius \(R_1\) is:
\[ R_1 = \frac{\sqrt{2qV}}{qB} \cdot \sqrt{m_X} \]
For particle \(Y\), the radius \(R_2\) is:
\[ R_2 = \frac{\sqrt{2qV}}{qB} \cdot \sqrt{m_Y} \]

5. **Finding the Ratio of Radii**:
Taking the ratio of the two radii:
\[ \frac{R_1}{R_2} = \frac{\sqrt{m_X}}{\sqrt{m_Y}} \]
Squaring both sides gives:
\[ \left(\frac{R_1}{R_2}\right)^2 = \frac{m_X}{m_Y} \]

6. **Final Result**:
Therefore, the ratio of the mass of particle \(X\) to the mass of particle \(Y\) is:
\[ \frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2 \]

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