A charged particle P leaves the origin with speed `v=v_0` at some inclination with the x-axis. There is a uniform magnetic field B along the x-axis. P strikes a fixed target T on the x-axis for a minimum value of `B=B_0`. P will also strike T if

Options

A. `B = 2B_(0), v = 2v_(0)`
B. `B = 2B_(0), v = v_(0)`
C. Both are correct
D. Both are wrong

Answer

To solve the problem, we need to analyze the motion of the charged particle \( P \) in a magnetic field. The particle leaves the origin with a speed \( v = v_0 \) at an angle \( \theta \) with the x-axis, and it is subjected to a uniform magnetic field \( B \) along the x-axis. We want to determine the conditions under which the particle strikes a fixed target \( T \) on the x-axis.

### Step-by-Step Solution:

1. **Understanding the Motion of the Charged Particle**:
The charged particle \( P \) experiences a magnetic force due to the magnetic field \( B \). The magnetic force \( F \) on a charged particle is given by:
\[ F = q(\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the particle, \( \mathbf{v} \) is its velocity, and \( \mathbf{B} \) is the magnetic field.

2. **Determining the Components of Velocity**:
The velocity \( \mathbf{v} \) can be resolved into two components:
- \( v_x = v_0 \cos \theta \) (along the x-axis)
- \( v_y = v_0 \sin \theta \) (along the y-axis)

3. **Calculating the Magnetic Force**:
Since the magnetic field \( \mathbf{B} \) is along the x-axis, the magnetic force will act in the y-direction:
\[ F_y = q(v_y B) = q(v_0 \sin \theta B) \]

4. **Finding the Radius of the Circular Motion**:
The particle will undergo circular motion in the y-z plane due to the magnetic force. The radius \( r \) of the circular motion can be expressed as:
\[ r = \frac{mv_y}{qB} = \frac{m(v_0 \sin \theta)}{qB} \]

5. **Determining the Pitch of the Helix**:
The pitch \( P \) of the helical path is the distance traveled along the x-axis in one complete revolution. The pitch is given by:
\[ P = \frac{2\pi m}{qB} v_x = \frac{2\pi m}{qB} (v_0 \cos \theta) \]

6. **Setting the Condition for Striking the Target**:
For the particle \( P \) to strike the target \( T \) on the x-axis, the pitch must be such that the particle returns to the x-axis after traveling a distance \( d \). The minimum value of \( B \) for which this occurs is \( B = B_0 \). Therefore, we can express the condition as:
\[ P = k \cdot B_0 \]
where \( k \) is a constant.

7. **Finding Additional Conditions**:
The particle will also strike the target if the magnetic field is increased. If we set \( B = 2B_0 \), we can find the new pitch:
\[ P' = \frac{2\pi m}{q(2B_0)} (v_0 \cos \theta) = \frac{P}{2} \]
This implies that the particle will still strike the target if \( B \) is doubled.

### Conclusion:
The charged particle \( P \) will strike the target \( T \) if:
1. The magnetic field \( B = B_0 \).
2. The magnetic field \( B = 2B_0 \).