A proton moving with a constant velocity passes through a region of space without any changing its velocity. If `E` and `B` represent the electric and magnetic fields, respectively. Then, this region of space may have

Options

A. E = 0, B = 0
B. `E = 0, B ne 0`
C. `E ne 0, B = 0`
D. `E ne 0, B ne 0`

Answer

To solve the problem, we need to analyze the conditions under which a proton moving with a constant velocity can pass through a region of space without changing its velocity.

### Step-by-Step Solution:

1. **Understanding the Forces on the Proton**:
- A proton is a positively charged particle. When it moves through an electric field (E) and a magnetic field (B), it experiences forces due to both fields.
- The force due to the electric field (F_E) is given by:
\[ F_E = qE \]
where \( q \) is the charge of the proton.

2. **Magnetic Force on the Proton**:
- The magnetic force (F_B) acting on a moving charge in a magnetic field is given by:
\[ F_B = q(v \times B) \]
- For a proton moving with velocity \( v \) perpendicular to the magnetic field \( B \), the magnitude of the magnetic force is:
\[ F_B = qvB \sin(90^\circ) = qvB \]

3. **Condition for Constant Velocity**:
- For the proton to maintain a constant velocity, the net force acting on it must be zero. This means the electric force must balance the magnetic force:
\[ F_E = F_B \]
- Substituting the expressions for the forces, we have:
\[ qE = qvB \]

4. **Simplifying the Equation**:
- Since the charge \( q \) is non-zero (for a proton), we can divide both sides by \( q \):
\[ E = vB \]

5. **Conclusion**:
- The proton can move with constant velocity in a region where both electric field \( E \) and magnetic field \( B \) are present, and they are related by the equation \( E = vB \). This implies that neither the electric field nor the magnetic field can be zero; both must exist and be perpendicular to each other and to the direction of motion of the proton.

### Final Answer:
The region of space may have both an electric field \( E \) and a magnetic field \( B \), such that \( E = vB \), where \( v \) is the velocity of the proton.