Let `vecE and vecB` denote the electric and magnetic field 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 statement are consistent with the observation.

Options

A. `barE=0` and `barB=0`
B. `barE ne 0` and `barB=0`
C. `vecE ne 0` and `vecB ne 0` and both `vecE` and `vecB` are parallel to `vecv`
D. `vecE` is parallel to `vecv` but `vecB` is perpendicular to `vecv`

Explanation

To solve the problem, we need to analyze the conditions under which a proton moving with a certain velocity can pass through a region with electric and magnetic fields without being deflected.

### Step-by-Step Solution:

1. **Understanding Forces on the Proton**:
A charged particle like a proton experiences two types of forces when moving through electric and magnetic fields:
- Electric Force (\( \vec{F}_E \)): Given by \( \vec{F}_E = q \vec{E} \), where \( q \) is the charge of the proton and \( \vec{E} \) is the electric field.
- Magnetic Force (\( \vec{F}_B \)): Given by \( \vec{F}_B = q (\vec{v} \times \vec{B}) \), where \( \vec{v} \) is the velocity of the proton and \( \vec{B} \) is the magnetic field.

2. **Condition for No Deflection**:
For the proton to pass through undeflected, the net force acting on it must be zero. This can happen under the following conditions:
- Both electric and magnetic fields are zero: \( \vec{E} = 0 \) and \( \vec{B} = 0 \).
- Electric field is present, but it is parallel to the velocity of the proton, and the magnetic field is zero: \( \vec{E} \neq 0 \) and \( \vec{B} = 0 \) (and \( \vec{E} \parallel \vec{v} \)).
- Both electric and magnetic fields are present, but both are parallel to the velocity of the proton: \( \vec{E} \neq 0 \), \( \vec{B} \neq 0 \), and \( \vec{E} \parallel \vec{v} \) and \( \vec{B} \parallel \vec{v} \).

3. **Evaluating the Options**:
- **Option 1**: \( \vec{E} = 0 \) and \( \vec{B} = 0 \) - This is valid as there are no forces acting on the proton.
- **Option 2**: \( \vec{E} \neq 0 \) and \( \vec{B} = 0 \) - This is valid only if \( \vec{E} \) is parallel to \( \vec{v} \). Since the option does not specify that, we cannot consider this option valid without additional information.
- **Option 3**: \( \vec{E} \neq 0 \), \( \vec{B} \neq 0 \), and both are parallel to \( \vec{v} \) - This is valid as both forces will not cause deflection.
- **Option 4**: \( \vec{E} \parallel \vec{v} \) and \( \vec{B} \perp \vec{v} \) - This is invalid because the magnetic force would act on the proton, causing it to deviate from its path.

4. **Conclusion**:
The correct options that are consistent with the observation of the proton passing undeflected are:
- Option 1: \( \vec{E} = 0 \) and \( \vec{B} = 0 \).
- Option 3: \( \vec{E} \neq 0 \), \( \vec{B} \neq 0 \), and both are parallel to \( \vec{v} \).