Two particles of masses `m_a` and `m_b` and same charge are projected in a perpendicular magnetic field . They travel along circular paths of radius `r_(a)` and `r_(b)` such that `r_(a) gt r_(b)` . Then which is true ?

Options

A. `m_a v_a gt m_bv_b`
B. `m_a gt m_b and v_a gt v_b`
C. `m_a = m_b and v_a = v_b`
D. `m_b v_b gt m_a v_a`

Answer

To solve the problem, let's analyze the motion of two charged particles in a magnetic field and derive the relationship between their masses and velocities based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Forces**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as the centripetal force required for circular motion. The magnetic force \( F_B \) is given by:
\[ F_B = qvB \]
where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the magnetic field strength.

2. **Centripetal Force**:
The centripetal force \( F_C \) required to keep a particle moving in a circular path of radius \( R \) is given by:
\[ F_C = \frac{mv^2}{R} \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

3. **Equating Forces**:
For a particle to move in a circular path in a magnetic field, the magnetic force must equal the centripetal force:
\[ qvB = \frac{mv^2}{R} \]

4. **Rearranging the Equation**:
We can rearrange this equation to express the radius \( R \) in terms of mass \( m \) and velocity \( v \):
\[ R = \frac{mv}{qB} \]

5. **Applying to Both Particles**:
For particle A with mass \( m_a \) and radius \( r_a \):
\[ r_a = \frac{m_a v_a}{qB} \]
For particle B with mass \( m_b \) and radius \( r_b \):
\[ r_b = \frac{m_b v_b}{qB} \]

6. **Given Condition**:
We know that \( r_a > r_b \). Therefore:
\[ \frac{m_a v_a}{qB} > \frac{m_b v_b}{qB} \]
Since \( q \) and \( B \) are the same for both particles, we can cancel them out:
\[ m_a v_a > m_b v_b \]

7. **Conclusion**:
This implies that the product of mass and velocity for particle A is greater than that for particle B. Thus, the correct relationship is:
\[ m_a v_a > m_b v_b \]

### Final Answer:
The correct option is that \( m_a v_a > m_b v_b \).