Two charged particles traverse identical helical paths in a completely opposite sense in a uniform magnetic field `vec(B)=B_(0)hat(K)`

Options

A. They have equal `z-`components of momenta.
B. They must have equal charges.
C. They necessarily represent a particle-antiparticle pair.
D. The charge to mass ratio satisfy:
`((e)/(m))_(1)+((e)/(m))_(2)=0`.

Explanation

To solve the problem regarding two charged particles traversing identical helical paths in opposite senses in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two charged particles moving in a uniform magnetic field \( \vec{B} = B_0 \hat{k} \).
- They are moving in helical paths but in opposite directions.
- They have equal z-components of momentum.

2. **Identifying the Key Parameters**:
- Let the charges of the particles be \( q_1 \) and \( q_2 \), and their masses be \( m_1 \) and \( m_2 \).
- Since they are moving in opposite directions but have the same z-component of momentum, we can denote the z-component of momentum as \( p_z = m v_z \), where \( v_z \) is the velocity component along the z-axis.

3. **Using the Helical Motion Concept**:
- The motion of a charged particle in a magnetic field results in a helical trajectory. The pitch \( p \) of the helix is given by:
\[ p = T v_z \]
where \( T \) is the period of the motion.

4. **Relating Charge and Mass**:
- The period \( T \) for a charged particle in a magnetic field is given by:
\[ T = \frac{2\pi m}{|q|B} \]
- Therefore, the pitch can be expressed as:
\[ p = \frac{2\pi m}{|q|B} v_z \]

5. **Setting Up the Equation**:
- For both particles, since they have identical helical paths, we can write:
\[ p_1 = p_2 \]
- This leads to:
\[ \frac{2\pi m_1}{|q_1|B} v_{z1} = \frac{2\pi m_2}{|q_2|B} v_{z2} \]
- Since \( v_{z1} = v_{z2} \) (equal z-components of momentum), we can simplify:
\[ \frac{m_1}{|q_1|} = \frac{m_2}{|q_2|} \]

6. **Considering the Opposite Directions**:
- Since the particles are moving in opposite senses, we can denote \( q_2 = -q_1 \).
- Thus, we have:
\[ \frac{m_1}{|q_1|} + \frac{m_2}{|q_2|} = 0 \]
- Substituting \( q_2 = -q_1 \):
\[ \frac{m_1}{|q_1|} - \frac{m_2}{|q_1|} = 0 \]

7. **Conclusion**:
- From the above equation, we conclude that:
\[ \frac{q_1}{m_1} + \frac{q_2}{m_2} = 0 \]
- This indicates that the two particles have equal and opposite charge-to-mass ratios, confirming that they represent a particle-antiparticle pair.

### Final Answer:
Thus, the solution shows that the two charged particles must have equal charges and represent a particle-antiparticle pair, leading to the conclusion that the charge-to-mass ratio satisfies the equation \( \frac{q_1}{m_1} + \frac{q_2}{m_2} = 0 \).