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. a. They have equal z-components of moment
B. b. They must have equal charges
C. c. They necessarily represent a particles, anti-particle pair
D. d. The charge to mass ratio satisfy
`((e)/(m))_(1) + ((e)/(m))_(2) = 0`

Answer

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

### Step-by-Step Solution:

1. **Understand the Motion of Charged Particles in a Magnetic Field**:
- When a charged particle moves in a magnetic field, it experiences a magnetic force that causes it to move in a circular path. If the particle also has a component of velocity parallel to the magnetic field, it will follow a helical path.

2. **Identify the Parameters**:
- Let the charge of the first particle be \( q_1 \) and its mass be \( m_1 \).
- Let the charge of the second particle be \( q_2 \) and its mass be \( m_2 \).
- Both particles are moving in a uniform magnetic field \( \vec{B} = B_0 \hat{k} \).

3. **Use the Formula for Helical Motion**:
- The pitch of the helical path can be expressed as:
\[ P = \frac{2 \pi m v \cos(\theta)}{q B} \]
- Here, \( P \) is the pitch, \( v \) is the speed of the particle, \( \theta \) is the angle of the velocity with respect to the magnetic field, and \( B \) is the magnetic field strength.

4. **Relate the Charges and Masses**:
- Since both particles traverse identical helical paths but in opposite directions, we can equate the ratios of charge to mass for both particles:
\[ \frac{q_1}{m_1} = -\frac{q_2}{m_2} \]
- This indicates that the charge-to-mass ratios are equal in magnitude but opposite in sign.

5. **Express the Condition Mathematically**:
- From the above relationship, we can write:
\[ \frac{q_1}{m_1} + \frac{q_2}{m_2} = 0 \]
- This equation shows that the sum of the charge-to-mass ratios of the two particles is zero.

6. **Conclusion**:
- Therefore, the correct option from the given choices is that the charge-to-mass ratio satisfies the equation:
\[ \frac{q_1}{m_1} + \frac{q_2}{m_2} = 0 \]

### Final Answer:
The correct option is D: The charge to mass ratio satisfies the equation \( \frac{q_1}{m_1} + \frac{q_2}{m_2} = 0 \).

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