Let $\overrightarrow E $ and $\overrightarrow B $ denote the electric and magnetic fields in a certain region of space. A proton moving with a velocity along a straight line enters the region and is found to pass through it undeflected. Indicate which of the following statements are consistent with the observations
A. $\overrightarrow E = 0$ and $\overrightarrow B = 0$
B. $\overrightarrow E \ne 0$ and $\overrightarrow B = 0$
C. $\overrightarrow E \ne 0$ and $\overrightarrow B \ne 0$ and both $\overrightarrow E $ and $\overrightarrow B $ are parallel to $\overrightarrow v $
D. $\overrightarrow E $ is parallel to $\overrightarrow v $ but $\overrightarrow B $ is perpendicular to $\overrightarrow v $

Hint: A charged particle will be deflected by magnetic field as well as electric field if its velocity is not parallel to the direction of electric field as well as magnetic field. A charged particle is deflected by both the magnetic field as well as the electric field.

Complete step by step answer:
From the hint, it is clear that a particle in an electric or magnetic field is deflected under the following conditions.
If the velocity of the particle is not parallel to both the fields.
Let’s discuss each option and check whether which options are correct.
When both $\overrightarrow E $ and $\overrightarrow B $ are zero, the particle will not be deflected. It is the same as having no fields at all.
Hence option A is correct.

When $\overrightarrow E \ne 0$ and $\overrightarrow B = 0$ , the particle will move in the direction of $\overrightarrow E $ and the particle can remain undeflected provided $\overrightarrow E ||\overrightarrow v $ .
Therefore, option B is also correct.

$\overrightarrow E \ne 0$ and $\overrightarrow B \ne 0$ and both $\overrightarrow E $ and $\overrightarrow B $ are parallel to $\overrightarrow v $ . Again, there will be no deflection due to the magnetic field and the particle will accelerate or decelerate in the direction of $\overrightarrow v $ .
Therefore, option C is also correct.

$\overrightarrow E $ is parallel to $\overrightarrow v $ but $\overrightarrow B $ is perpendicular to $\overrightarrow v $ . In this case, the particle will be deflected as $\overrightarrow B $ is perpendicular to $\overrightarrow v $ and not parallel.
Therefore, option D is incorrect.

Therefore, option A,B and C are the correct options.

Note:
Remember that particles will be undeflected if the magnitude of both the fields is simultaneously zero. Also, the particle will be undeflected if its velocity is parallel to the fields. Remember that if the particle has velocity at some angle other than zero, then the particle will be deflected.