1. Condition for no change in velocity

For a charged particle moving in electric and magnetic fields, the Lorentz force is

If the proton passes through the region without any change in its velocity, then its acceleration is zero, so the net force must be zero:

So we check each case.

2. Case (A): $E=0,\; B=0$

Then

So the proton can move with constant velocity.

✅ Possible

3. Case (B): $E=0,\; B\neq 0$

Then

This can be zero if

(or antiparallel), because then

Hence constant velocity is possible.

✅ Possible

4. Case (C): $E\neq 0,\; B=0$

Then

Since $\vec{E} \neq 0$, the force is nonzero, so the proton must accelerate and its velocity will change.

❌ Not possible

5. Case (D): $E\neq 0,\; B\neq 0$

Then it is possible that

so that the net force becomes zero:

Thus the proton can still move with unchanged velocity.

✅ Possible

6. Final selection

Possible cases are:

• (A)
• (B)
• (D)

So the correct option is

7. Comparison with stored answer

Stored correct answer: C

Derived answer: C

They agree.