Two particles X and Y having equal charge , after being accelerated through the same potential difference entre a region of uniform magnetic field and describe circular paths of radii `R_1 and R_2` respectively. The ration of the mass of X to that of Y is

Options

A. `sqrt(R_1/(R_2))`
B. `(R_2/R_1)^2`
C. `(R_1/R_2)^2`
D. `sqrt(R_2/R_1)`

Explanation

To solve the problem, we need to find the ratio of the mass of particle X to that of particle Y, given that both particles have equal charge and are accelerated through the same potential difference in a uniform magnetic field, resulting in circular paths of radii \( R_1 \) and \( R_2 \) respectively.

### Step-by-Step Solution:

1. **Understanding the Forces in Circular Motion**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as the centripetal force required to keep it in circular motion. The magnetic force \( F \) on a charged particle moving with velocity \( v \) in a magnetic field \( B \) is given by:
\[ F = Bqv \]
where \( q \) is the charge of the particle.

2. **Centripetal Force**:
The centripetal force required to keep a particle moving in a circle of radius \( R \) is given by:
\[ F = \frac{mv^2}{R} \]
where \( m \) is the mass of the particle.

3. **Setting the Forces Equal**:
Since the magnetic force provides the centripetal force, we can set these two equations equal:
\[ Bqv = \frac{mv^2}{R} \]

4. **Rearranging the Equation**:
We can rearrange this equation to express \( R \) in terms of \( m \):
\[ R = \frac{mv}{Bq} \]

5. **Finding Velocity**:
The kinetic energy gained by the particle when it is accelerated through a potential difference \( V \) is given by:
\[ KE = \frac{1}{2} mv^2 = qV \]
From this, we can solve for \( v \):
\[ v = \sqrt{\frac{2qV}{m}} \]

6. **Substituting for Velocity**:
Substitute \( v \) back into the equation for \( R \):
\[ R = \frac{m \sqrt{\frac{2qV}{m}}}{Bq} = \frac{\sqrt{2qV} \cdot \sqrt{m}}{Bq} \]
Simplifying gives:
\[ R = \frac{\sqrt{2V}}{B} \cdot \sqrt{\frac{m}{q}} \]

7. **Relating the Radii**:
For particles X and Y, we have:
\[ R_1 = \frac{\sqrt{2V}}{B} \cdot \sqrt{\frac{m_X}{q}} \quad \text{and} \quad R_2 = \frac{\sqrt{2V}}{B} \cdot \sqrt{\frac{m_Y}{q}} \]
Since both particles have the same charge \( q \) and are accelerated through the same potential difference \( V \), we can write:
\[ \frac{R_1}{R_2} = \frac{\sqrt{m_X}}{\sqrt{m_Y}} \]

8. **Squaring Both Sides**:
Squaring both sides gives:
\[ \left(\frac{R_1}{R_2}\right)^2 = \frac{m_X}{m_Y} \]

9. **Finding the Mass Ratio**:
Thus, the ratio of the masses is:
\[ \frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2 \]

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