A proton enters in a uniform electric and magnetic fields E and B respectively. Velocity of proton is v. All the three vectors are mutually perpendicular. The proton is deflected along positive x-axis when either of the fields or both are switched on simultaneously. Which of the following statement(s) is/are correct ?

Options

A. V may be along positive y-axis
B. E is along positive x-axis
C. B may be along positive z-axis
D. B may be along negative y-axis

Answer

To solve the problem, we need to analyze the situation involving a proton moving through uniform electric and magnetic fields, where all vectors are mutually perpendicular. We will determine the correctness of the statements based on the forces acting on the proton.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The proton is moving with velocity \( \vec{v} \).
- The electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are both present.
- All three vectors \( \vec{E} \), \( \vec{B} \), and \( \vec{v} \) are mutually perpendicular.

2. **Direction of Forces**:
- The force on the proton due to the electric field is given by \( \vec{F_E} = q\vec{E} \), where \( q \) is the charge of the proton.
- The force on the proton due to the magnetic field is given by \( \vec{F_B} = q(\vec{v} \times \vec{B}) \).

3. **Deflection along the x-axis**:
- The problem states that the proton is deflected along the positive x-axis when either or both fields are switched on.
- This means that the net force acting on the proton must have a component in the x-direction.

4. **Analyzing the Electric Field**:
- For the proton to be deflected in the positive x-direction, the electric field \( \vec{E} \) must also point in the positive x-direction. This is because the electric force will act in the direction of the electric field.

5. **Analyzing the Magnetic Field**:
- Since \( \vec{E} \) is along the x-axis and \( \vec{v} \) is perpendicular to \( \vec{E} \), we can assume \( \vec{v} \) is along the y-axis (either positive or negative).
- The magnetic field \( \vec{B} \) must then be along the z-axis (either positive or negative) to maintain the mutual perpendicularity.

6. **Evaluating the Statements**:
- **Statement 1**: "V may be along positive y-axis." - This is correct because if \( \vec{v} \) is along the positive y-axis, the electric field \( \vec{E} \) can still deflect the proton in the x-direction.
- **Statement 2**: "E may be along positive x-axis." - This is correct as established earlier.
- **Statement 3**: "B may be along positive z-direction." - This is also correct as it maintains the perpendicularity with \( \vec{E} \) and \( \vec{v} \).
- **Statement 4**: "V may be along negative y-axis." - This is incorrect because if \( \vec{v} \) is along the negative y-axis and \( \vec{B} \) is also along the negative y-axis, the magnetic force would be zero (as \( \sin(180^\circ) = 0 \)), and the proton would not be deflected.

### Conclusion:
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Correct
- Statement 4: Incorrect