In a certain region of space electric field `vecE` and magnetic field `vecB` are perpendicular to each other and an electron enters in region perpendicular to the direction of `vecB` and `vecE` both and move underflected, then velocity of electron is

Options

A. `(abs(vecE))/(abs(vecB))`
B. `(abs(vecB))/(abs(vecE))`
C. `vecExxvecB`
D. `vecE.vecB`

Answer

To solve the problem, we need to analyze the forces acting on the electron in the presence of both electric and magnetic fields. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Forces**:
The forces acting on a charged particle (like an electron) in an electric field \( \vec{E} \) and a magnetic field \( \vec{B} \) are given by:
\[ \vec{F} = q\vec{E} + q(\vec{V} \times \vec{B}) \]
where \( q \) is the charge of the electron, \( \vec{V} \) is the velocity of the electron, and \( \times \) denotes the cross product.

2. **Condition for Undeflected Motion**:
The problem states that the electron moves undeflected. This means that the net force acting on it is zero:
\[ \vec{F} = 0 \]
Therefore, we can write:
\[ q\vec{E} + q(\vec{V} \times \vec{B}) = 0 \]
Since the charge \( q \) (for an electron, \( q = -e \)) is non-zero, we can simplify this to:
\[ \vec{E} + \vec{V} \times \vec{B} = 0 \]

3. **Rearranging the Equation**:
From the equation \( \vec{E} + \vec{V} \times \vec{B} = 0 \), we can rearrange it to find:
\[ \vec{E} = -(\vec{V} \times \vec{B}) \]

4. **Magnitude of the Fields**:
Taking magnitudes on both sides, we have:
\[ |\vec{E}| = |\vec{V} \times \vec{B}| \]
The magnitude of the cross product can be expressed as:
\[ |\vec{V} \times \vec{B}| = |\vec{V}||\vec{B}|\sin(\theta) \]
where \( \theta \) is the angle between \( \vec{V} \) and \( \vec{B} \). Since the electron enters perpendicular to both \( \vec{E} \) and \( \vec{B} \), \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1 \). Thus, we can simplify to:
\[ |\vec{E}| = |\vec{V}||\vec{B}| \]

5. **Finding the Velocity**:
Rearranging the above equation gives us:
\[ |\vec{V}| = \frac{|\vec{E}|}{|\vec{B}|} \]
This is the expression for the velocity of the electron when it moves undeflected in the presence of perpendicular electric and magnetic fields.

### Final Answer:
The velocity of the electron is given by:
\[ |\vec{V}| = \frac{|\vec{E}|}{|\vec{B}|} \]