Two particles A and B having equal charges `+6C`, after being accelerated through the same potential differences, enter a region of uniform magnetic field and describe circular paths of radii `2cm` and `3cm` respectively. The ratio of mass of A to that of B is

Options

A. `1//3`
B. `1//2`
C. `4//9`
D. `9//5`

Answer

To solve the problem, we need to find the ratio of the masses of particles A and B based on the information given about their charges, potential differences, and the radii of their circular paths in a magnetic field.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Charges of both particles: \( Q_A = Q_B = +6 \, C \)
- Radii of circular paths: \( R_A = 2 \, cm = 0.02 \, m \) and \( R_B = 3 \, cm = 0.03 \, m \)
- Both particles are accelerated through the same potential difference.

2. **Using 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 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.

3. **Relating Velocity to Potential Difference**:
When a charged particle is accelerated through a potential difference \( V \), its kinetic energy \( K.E. \) is given by:
\[ K.E. = QV \]
This kinetic energy can also be expressed in terms of mass and velocity:
\[ K.E. = \frac{1}{2} mv^2 \]
Setting these equal gives:
\[ QV = \frac{1}{2} mv^2 \]
Rearranging for \( v \):
\[ v = \sqrt{\frac{2QV}{m}} \]

4. **Substituting Velocity in the Radius Formula**:
Substitute \( v \) into the radius formula:
\[ R = \frac{m}{QB} \sqrt{\frac{2QV}{m}} = \frac{1}{QB} \sqrt{2QVm} \]
This implies:
\[ R^2 = \frac{2QV}{Q^2B^2} m \]
Thus, we can express the ratio of the masses based on the radii:
\[ \frac{R_A^2}{R_B^2} = \frac{m_A}{m_B} \]

5. **Calculating the Ratio of Masses**:
Now, substituting the values of the radii:
\[ R_A = 0.02 \, m, \quad R_B = 0.03 \, m \]
\[ \frac{R_A^2}{R_B^2} = \frac{(0.02)^2}{(0.03)^2} = \frac{0.0004}{0.0009} = \frac{4}{9} \]
Therefore, the ratio of the masses is:
\[ \frac{m_A}{m_B} = \frac{4}{9} \]

### Final Answer:
The ratio of the mass of particle A to that of particle B is:
\[ \frac{m_A}{m_B} = \frac{4}{9} \]