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.

Let's analyze the conditions under which a proton moves with a constant velocity $v_0$ (i.e., its velocity does not change, meaning the net force acting on it is zero). The net force on a charged particle in the presence of both electric and magnetic fields is given by the Lorentz force formula:
$F=q(E+v\times B)$
For the velocity of the proton to remain unchanged, the acceleration must be zero, which means the net Lorentz force must be zero:
$F=0\Rightarrow E+v_0\times B=0$

Now, let's evaluate each option to see if $F=0$ can be satisfied:

(a) E = 0, B = 0:
If both the electric field $E$ and the magnetic field $B$ are zero, the net force is:
$F=q(0+v_0\times 0)=0$
This is a valid condition, so (a) is true.

(b) E = 0, B ≠ 0:
If $E=0$ and $B\ne 0$, the net force is:
$F=q(v_0\times B)$
For this force to be zero, the cross product $v_0\times B$ must be zero. This occurs when the velocity vector $v_0$ is parallel or antiparallel to the magnetic field vector $B$ (i.e., the angle between them is 0° or 180°). Thus, it is possible for the force to be zero. Therefore, (b) may be true.

(c) E ≠ 0, B = 0:
If $E\ne 0$ and $B=0$, the net force is:
$F=qE$
Since $E\ne 0$ and $q\ne 0$ (as a proton has a non-zero charge), the net force $F$ cannot be zero. Thus, the velocity must change, meaning (c) cannot be true.

(d) E ≠ 0, B ≠ 0:
If both $E\ne 0$ and $B\ne 0$, we can achieve a net zero force if the electric force and the magnetic force are equal in magnitude and opposite in direction:
$qE=-q(v_0\times B)\Rightarrow E=-(v_0\times B)$
This is the principle of a velocity selector, where the electric and magnetic fields are perpendicular to each other and to the velocity of the particle, such that $E=v_{0}B$. Under this configuration, the net force is zero and the velocity remains constant. Therefore, (d) may be true.

Thus, the conditions that may be true are (a), (b), and (d).