Two charged particles have charges and masses in the ratio `2:3` and `1:4` respectively. If they enter a uniform magnetic field and move with the same velocity then the ratio of their respective time periods of revolutions is

Options

A. `3:8`
B. `1:4`
C. `3:5`
D. `1:6`

Answer

To find the ratio of the time periods of revolution of two charged particles moving in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Time Period Formula:**
The time period \( T \) of a charged particle moving in a magnetic field can be expressed as:
\[ T = \frac{2\pi r}{v} \]
where \( r \) is the radius of the circular path and \( v \) is the velocity of the particle.

2. **Setting Up the Ratio of Time Periods:**
For two particles, the ratio of their time periods \( T_1 \) and \( T_2 \) can be written as:
\[ \frac{T_1}{T_2} = \frac{r_1}{v_1} \cdot \frac{v_2}{r_2} \]

3. **Equating Centripetal Force and Magnetic Force:**
The magnetic force acting on a charged particle is given by:
\[ F = QvB \]
This magnetic force acts as the centripetal force, which is given by:
\[ F = \frac{mv^2}{r} \]
Thus, we can equate the two forces:
\[ QvB = \frac{mv^2}{r} \]

4. **Rearranging the Equation:**
From the above equation, we can derive:
\[ \frac{v}{r} = \frac{QB}{m} \]

5. **Substituting into the Time Period Ratio:**
Now, substituting \( \frac{v}{r} \) back into the time period ratio:
\[ \frac{T_1}{T_2} = \frac{m_1}{Q_1B} \cdot \frac{Q_2B}{m_2} \]
This simplifies to:
\[ \frac{T_1}{T_2} = \frac{m_1}{m_2} \cdot \frac{Q_2}{Q_1} \]

6. **Using Given Ratios:**
We are given the ratios of masses and charges:
- Mass ratio \( \frac{m_1}{m_2} = \frac{1}{4} \)
- Charge ratio \( \frac{Q_1}{Q_2} = \frac{2}{3} \)

Therefore, we can express \( \frac{Q_2}{Q_1} \) as:
\[ \frac{Q_2}{Q_1} = \frac{3}{2} \]

7. **Calculating the Final Ratio:**
Now substituting these values into the time period ratio:
\[ \frac{T_1}{T_2} = \frac{1/4}{1} \cdot \frac{3/2}{1} = \frac{1}{4} \cdot \frac{3}{2} = \frac{3}{8} \]

### Final Answer:
Thus, the ratio of their respective time periods of revolutions is:
\[ \frac{T_1}{T_2} = \frac{3}{8} \]