Two particle X and Y having equal charge, after being accelerated through 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_1/R_2`
C. `(R_1/R_2)^2`
D. `(R_1)(R_2)`

Answer

To solve the problem step by step, we will analyze the given information and apply relevant physics principles.

### Step 1: Understand the relationship between radius, mass, velocity, and charge in a magnetic field.
When a charged particle moves in a magnetic field, it experiences a centripetal force that causes it to move in a circular path. The radius \( R \) of the circular path is given by the formula:

\[ R = \frac{mv}{qB} \]

where:
- \( R \) is the radius of the circular path,
- \( m \) 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.

### Step 2: Write the equations for both particles X and Y.
For particle X with radius \( R_1 \):

\[ R_1 = \frac{m_X v_X}{qB} \]

For particle Y with radius \( R_2 \):

\[ R_2 = \frac{m_Y v_Y}{qB} \]

### Step 3: Relate the kinetic energy to the potential difference.
Since both particles are accelerated through the same potential difference \( V \), their kinetic energies can be expressed as:

\[ \frac{1}{2} m_X v_X^2 = qV \]
\[ \frac{1}{2} m_Y v_Y^2 = qV \]

From these equations, we can express the velocities in terms of mass and charge:

\[ v_X = \sqrt{\frac{2qV}{m_X}} \]
\[ v_Y = \sqrt{\frac{2qV}{m_Y}} \]

### Step 4: Substitute the velocities back into the radius equations.
Substituting \( v_X \) and \( v_Y \) into the equations for \( R_1 \) and \( R_2 \):

For particle X:

\[ R_1 = \frac{m_X \sqrt{\frac{2qV}{m_X}}}{qB} = \frac{\sqrt{2m_X qV}}{qB} \]

For particle Y:

\[ R_2 = \frac{m_Y \sqrt{\frac{2qV}{m_Y}}}{qB} = \frac{\sqrt{2m_Y qV}}{qB} \]

### Step 5: Square both equations to eliminate the square root.
Squaring both equations gives:

\[ R_1^2 = \frac{2m_X qV}{(qB)^2} \]
\[ R_2^2 = \frac{2m_Y qV}{(qB)^2} \]

### Step 6: Set up the ratio of the masses.
From the equations for \( R_1^2 \) and \( R_2^2 \), we can set up the ratio:

\[ \frac{R_1^2}{R_2^2} = \frac{m_X}{m_Y} \]

### Step 7: Rearrange to find the ratio of masses.
Thus, we can express the ratio of the masses as:

\[ \frac{m_X}{m_Y} = \frac{R_1^2}{R_2^2} \]

### Final Result:
The ratio of the mass of particle X to that of particle Y is:

\[ \frac{m_X}{m_Y} = \frac{R_1^2}{R_2^2} \]