Two particles X and Y having equal charges are being accelerated through the same potential difference. Thereafter they enter normally in a region of uniform magnetic field and describes circular paths of radii `R_1` and `R_2` respectively. The mass ratio of X and Y is :

Options

A. `(frac{R_2}{R_1})^2`
B. `(frac{R_1}{R_2})^2`
C. `(frac{R_1}{R_2})`
D. `(frac{R_2}{R_1})`

Answer

To solve the problem, we need to derive the relationship between the masses of the two particles X and Y based on the given information. Here’s a step-by-step solution:

### Step 1: Understand the scenario
We have two particles X and Y with equal charges (let's denote the charge as \( q \)) that are accelerated through the same potential difference \( V \). After acceleration, they enter a uniform magnetic field perpendicularly and describe circular paths with radii \( R_1 \) and \( R_2 \) respectively.

### Step 2: Relate kinetic energy to potential difference
When a charged particle is accelerated through a potential difference \( V \), it gains kinetic energy equal to the work done on it by the electric field:
\[ K.E. = qV \]
This kinetic energy can also be expressed in terms of mass and velocity:
\[ K.E. = \frac{1}{2} mv^2 \]
Equating the two expressions for kinetic energy, we have:
\[ \frac{1}{2} mv^2 = qV \]

### Step 3: Solve for velocity
From the equation above, we can solve for the velocity \( v \):
\[ v = \sqrt{\frac{2qV}{m}} \]

### Step 4: Use the formula for radius in a magnetic field
The radius \( R \) of the circular path of a charged particle moving in a magnetic field is given by:
\[ R = \frac{mv}{qB} \]
Substituting the expression for \( v \) from Step 3 into this equation, we get:
\[ R = \frac{m \sqrt{\frac{2qV}{m}}}{qB} \]
This simplifies to:
\[ R = \frac{\sqrt{2m qV}}{qB} \]

### Step 5: Express the relationship for both particles
For particle X with radius \( R_1 \) and mass \( m_1 \):
\[ R_1 = \frac{\sqrt{2m_1 qV}}{qB} \]
For particle Y with radius \( R_2 \) and mass \( m_2 \):
\[ R_2 = \frac{\sqrt{2m_2 qV}}{qB} \]

### Step 6: Set up the ratio of the radii
Dividing the two equations gives:
\[ \frac{R_1}{R_2} = \frac{\sqrt{2m_1 qV}}{\sqrt{2m_2 qV}} = \sqrt{\frac{m_1}{m_2}} \]

### Step 7: Square both sides to find the mass ratio
Squaring both sides results in:
\[ \left(\frac{R_1}{R_2}\right)^2 = \frac{m_1}{m_2} \]
Thus, the ratio of the masses is:
\[ \frac{m_1}{m_2} = \left(\frac{R_1}{R_2}\right)^2 \]

### Conclusion
The mass ratio of particles X and Y is given by:
\[ \frac{m_1}{m_2} = \frac{R_1^2}{R_2^2} \]