A proton having velocity V₀ passes through a region having electric field E and magnetic field B. If the velocity of proton does not change, then which of the following may be true?

(a) E = 0, B = 0

(b) E = 0, B ≠ 0

(c) E ≠ 0, B = 0

(d) E ≠ 0, B ≠ 0

The correct option is a, b, d.

To understand why this is correct, we can analyze the forces acting on the proton. A proton is a positively charged particle with charge q. When a charged particle moves through a region with an electric field E and a magnetic field B, it experiences the Lorentz force F, which is given by the formula:

$\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times \mathbf{B}\right)$

where v is the velocity of the proton.

For the velocity of the proton to remain completely unchanged, the net acceleration must be zero, which means the net force acting on the particle must be zero:

$\mathbf{F}=0$

Substituting this condition into the Lorentz force equation, we get:

$\mathbf{E}+\mathbf{v}\times \mathbf{B}=0$

Let us analyze each statement based on this condition:

Statement (a): E = 0, B = 0
If there is no electric field (E = 0) and no magnetic field (B = 0), the net force acting on the proton is zero. Consequently, the velocity of the proton remains constant. Therefore, statement (a) may be true.

Statement (b): E = 0, B ≠ 0
If the electric field is zero (E = 0) but the magnetic field is non-zero (B ≠ 0), the net force becomes:

$\mathbf{F}=q\left(\mathbf{v}\times \mathbf{B}\right)$

For the force to be zero, the cross product of the velocity and magnetic field vectors must be zero:

$\mathbf{v}\times \mathbf{B}=0$

This is satisfied if the proton's velocity is parallel or antiparallel to the direction of the magnetic field. In this case, the magnetic force is zero, and the velocity does not change. Therefore, statement (b) may be true.

Statement (c): E ≠ 0, B = 0
If there is a non-zero electric field (E ≠ 0) and no magnetic field (B = 0), the force on the proton is:

$\mathbf{F}=q\mathbf{E}$

Since the electric field is non-zero, a net force will act on the proton, which will accelerate it and change its velocity. Therefore, statement (c) cannot be true.

Statement (d): E ≠ 0, B ≠ 0
If both fields are non-zero, the net force is zero if the electric field balances the magnetic force:

$\mathbf{E}=-\left(\mathbf{v}\times \mathbf{B}\right)$

This condition can be satisfied if the electric field, the velocity, and the magnetic field are mutually perpendicular, and their magnitudes satisfy the relation $E=vB$ in such a way that the electric and magnetic forces act in opposite directions and cancel each other out (the velocity selector condition). Therefore, statement (d) may be true.

Comparing these conclusions, the statements that may be true are (a), (b), and (d).