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 where a proton enters a uniform electric field (E) and a magnetic field (B), with its velocity (v) such that all three vectors are mutually perpendicular. The proton is deflected along the positive x-axis when either of the fields or both are switched on simultaneously.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have three vectors: Electric field (E), Magnetic field (B), and Velocity (v) of the proton.
- All three vectors are mutually perpendicular to each other.
- Let's assume:
- E is along the positive x-axis.
- v is along the positive y-axis.
- B is along the positive z-axis.

2. **Effect of Electric Field (E)**:
- When only the electric field is switched on, the force (F) acting on the proton due to the electric field is given by:
\[ F_E = qE \]
where \( q \) is the charge of the proton.
- Since the electric field is along the positive x-axis, the proton will experience a force in the positive x-direction, causing it to deflect along the positive x-axis.

3. **Effect of Magnetic Field (B)**:
- When only the magnetic field is switched on, the magnetic force (F_M) acting on the proton is given by:
\[ F_M = q(v \times B) \]
- Here, v is in the positive y-direction and B is in the positive z-direction. The cross product \( v \times B \) will give a force in the negative x-direction (using the right-hand rule).
- Therefore, if only the magnetic field is switched on, the proton would deflect in the negative x-direction, which contradicts the given condition that it deflects along the positive x-axis.

4. **Simultaneous Effect of Both Fields**:
- When both fields are switched on, the total force on the proton will be the vector sum of the electric and magnetic forces.
- If the electric field is still in the positive x-direction and the magnetic field is in the positive z-direction, then:
- The electric force will push the proton in the positive x-direction.
- The magnetic force, calculated as \( q(v \times B) \), will still act in the negative x-direction if the velocity is in the positive y-direction.
- For the proton to continue moving in the positive x-direction, the forces must balance, i.e., \( F_E = F_M \).

5. **Conclusion**:
- The proton can only deflect along the positive x-axis if the electric field is acting in the positive x-direction and the magnetic field is in the positive z-direction.
- Therefore, the correct statements regarding the directions of the fields and their effects on the proton's motion can be summarized as follows:
- Electric field can be along the positive x-axis.
- Magnetic field can be along the positive z-axis.
- The proton deflects along the positive x-axis when the electric field is active.

### Final Answer:
Only the statement that the electric field is along the positive x-axis and the magnetic field is along the positive z-axis is correct.