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

Answer

To solve the problem, we need to analyze the motion of the two charged particles \(X\) and \(Y\) in a magnetic field after being accelerated through the same potential difference. Here is a step-by-step solution:

### Step 1: Understand the Kinetic Energy
When a charged particle is accelerated through a potential difference \(V\), it gains kinetic energy given by:
\[ KE = qV \]
where \(q\) is the charge of the particle.

### Step 2: Relate Kinetic Energy to Velocity
The kinetic energy can also be expressed in terms of mass \(m\) and velocity \(v\):
\[ KE = \frac{1}{2} mv^2 \]
Setting the two expressions for kinetic energy equal gives:
\[ qV = \frac{1}{2} mv^2 \]

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

### Step 4: Analyze the Motion in the Magnetic Field
When the particles enter a uniform magnetic field \(B\), they experience a magnetic force that acts as the centripetal force required for circular motion. The magnetic force \(F\) is given by:
\[ F = qvB \]
For circular motion, the centripetal force is given by:
\[ F = \frac{mv^2}{R} \]
where \(R\) is the radius of the circular path.

### Step 5: Set the Forces Equal
Setting the magnetic force equal to the centripetal force gives:
\[ qvB = \frac{mv^2}{R} \]

### Step 6: Rearrange to Find Radius
Rearranging the equation to solve for the radius \(R\) yields:
\[ R = \frac{mv}{qB} \]

### Step 7: Substitute for Velocity
Substituting the expression for \(v\) from Step 3 into the equation for \(R\):
\[ R = \frac{m \sqrt{\frac{2qV}{m}}}{qB} = \frac{\sqrt{2m qV}}{qB} \]

### Step 8: Relate the Radii of Particles X and Y
Let \(R_1\) be the radius for particle \(X\) and \(R_2\) for particle \(Y\). Since both particles have equal charge \(q\), are accelerated through the same potential \(V\), and are in the same magnetic field \(B\), we can write:
\[ R_1 = \frac{\sqrt{2m_X qV}}{qB} \quad \text{and} \quad R_2 = \frac{\sqrt{2m_Y qV}}{qB} \]

### Step 9: Form the Ratio of Radii
Taking the ratio of the two radii:
\[ \frac{R_1}{R_2} = \frac{\sqrt{m_X}}{\sqrt{m_Y}} \]

### Step 10: Square Both Sides
Squaring both sides gives:
\[ \left(\frac{R_1}{R_2}\right)^2 = \frac{m_X}{m_Y} \]

### Step 11: Conclusion
Thus, the ratio of the masses of particles \(X\) and \(Y\) is:
\[ \frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2 \]