If two charged particles of same mass and charge are describing circles in the same magnetic field with radii `r_(1)` and `r_(2)(gt r_(1))` the speed of the first particle is … that of the second particle while the time period of the particle is …that of the second partilcle.

Answer

To solve the problem, we need to analyze the motion of two charged particles moving in a magnetic field. Let's denote the two particles as Particle 1 and Particle 2, with radii \( r_1 \) and \( r_2 \) respectively, where \( r_2 > r_1 \).

### Step-by-Step Solution:

1. **Understanding the Forces**:
The magnetic force acting on a charged particle moving in a magnetic field provides the centripetal force necessary for circular motion. The magnetic force \( F \) on a charged particle is given by:
\[ F = qvB \]
where \( q \) is the charge, \( v \) is the speed of the particle, and \( B \) is the magnetic field strength.

2. **Centripetal Force**:
The centripetal force required to keep a particle moving in a circle of radius \( r \) is given by:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle.

3. **Equating Forces**:
Since the magnetic force acts as the centripetal force, we can set these two equations equal to each other for both particles:
\[ qv_1B = \frac{mv_1^2}{r_1} \quad \text{(for Particle 1)} \]
\[ qv_2B = \frac{mv_2^2}{r_2} \quad \text{(for Particle 2)} \]

4. **Solving for Speed**:
From the above equations, we can express the speeds in terms of the radii:
\[ v_1 = \frac{qBr_1}{m} \quad \text{and} \quad v_2 = \frac{qBr_2}{m} \]
Taking the ratio of the speeds:
\[ \frac{v_1}{v_2} = \frac{r_1}{r_2} \]

5. **Time Period Calculation**:
The time period \( T \) for circular motion is given by:
\[ T = \frac{2\pi r}{v} \]
For both particles, we can write:
\[ T_1 = \frac{2\pi r_1}{v_1} \quad \text{and} \quad T_2 = \frac{2\pi r_2}{v_2} \]

6. **Taking the Ratio of Time Periods**:
Now, substituting the expressions for \( v_1 \) and \( v_2 \):
\[ T_1 = \frac{2\pi r_1}{\frac{qBr_1}{m}} = \frac{2\pi m}{qB} \]
\[ T_2 = \frac{2\pi r_2}{\frac{qBr_2}{m}} = \frac{2\pi m}{qB} \]
Thus, the ratio of the time periods is:
\[ \frac{T_1}{T_2} = 1 \]

### Final Answers:
- The speed of the first particle is **proportional** to that of the second particle, specifically \( \frac{v_1}{v_2} = \frac{r_1}{r_2} \).
- The time period of the first particle is **equal** to that of the second particle, \( T_1 = T_2 \).