Two ions having masses in the ratio `1 : 1` and charges `1 : 2` are projected into uniform magnetic field perpendicular to the field with speeds in th ratio `2 : 3`. The ratio of the radius of circular paths along which the two particles move is

Options

A. `4:3`
B. `2:3`
C. `3:1`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of the radii of the circular paths of two ions projected into a uniform magnetic field. Let's denote the two ions as A and B.

### Given Data:
1. Mass ratio: \( \frac{m_A}{m_B} = \frac{1}{1} \) (or \( m_A = m_B \))
2. Charge ratio: \( \frac{Q_A}{Q_B} = \frac{1}{2} \) (or \( Q_B = 2Q_A \))
3. Velocity ratio: \( \frac{V_A}{V_B} = \frac{2}{3} \)

### Step-by-Step Solution:

1. **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:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity,
- \( q \) is the charge,
- \( B \) is the magnetic field strength.

2. **Calculate Radius for Ion A**:
For ion A, the radius \( r_A \) can be expressed as:
\[ r_A = \frac{m_A V_A}{Q_A B} \]

3. **Calculate Radius for Ion B**:
For ion B, the radius \( r_B \) can be expressed as:
\[ r_B = \frac{m_B V_B}{Q_B B} \]

4. **Finding the Ratio of Radii**:
To find the ratio \( \frac{r_A}{r_B} \):
\[ \frac{r_A}{r_B} = \frac{\frac{m_A V_A}{Q_A B}}{\frac{m_B V_B}{Q_B B}} = \frac{m_A V_A Q_B}{m_B V_B Q_A} \]
The \( B \) cancels out from the numerator and denominator.

5. **Substituting Known Ratios**:
Substitute the known ratios into the equation:
- Since \( \frac{m_A}{m_B} = 1 \), we have \( m_A = m_B \) (let's denote it as \( m \)).
- \( \frac{V_A}{V_B} = \frac{2}{3} \).
- Since \( \frac{Q_A}{Q_B} = \frac{1}{2} \), we have \( Q_B = 2Q_A \) or \( \frac{Q_B}{Q_A} = 2 \).

Therefore,
\[ \frac{r_A}{r_B} = \frac{m_A}{m_B} \cdot \frac{V_A}{V_B} \cdot \frac{Q_B}{Q_A} = 1 \cdot \frac{2}{3} \cdot 2 \]

6. **Calculating the Final Ratio**:
\[ \frac{r_A}{r_B} = \frac{2}{3} \cdot 2 = \frac{4}{3} \]

### Final Answer:
The ratio of the radius of circular paths along which the two particles move is:
\[ \frac{r_A}{r_B} = \frac{4}{3} \]