The accelertion of a electron at a certain moment in a magnetic field `B=2hati+3hatj+4hatk` is `a=xhati+hatj-hatk`. The value of `x` is

Options

A. `0.5`
B. `1`
C. `2.5`
D. `1.5`

Explanation

To solve the problem, we need to find the value of \( x \) in the acceleration vector \( \mathbf{a} = x \hat{i} + \hat{j} - \hat{k} \) given the magnetic field \( \mathbf{B} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \).

### Step-by-step Solution:

1. **Understanding the Relationship**:
The force acting on a charged particle (like an electron) in a magnetic field is given by the Lorentz force equation:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the electron, \( \mathbf{v} \) is the velocity, and \( \mathbf{B} \) is the magnetic field. The acceleration \( \mathbf{a} \) is related to the force by Newton's second law:
\[ \mathbf{F} = m \mathbf{a} \]

2. **Using the Perpendicularity Condition**:
Since the acceleration \( \mathbf{a} \) is perpendicular to the magnetic field \( \mathbf{B} \), we can use the dot product:
\[ \mathbf{a} \cdot \mathbf{B} = 0 \]

3. **Substituting the Vectors**:
Substitute the vectors into the dot product equation:
\[ (x \hat{i} + \hat{j} - \hat{k}) \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = 0 \]

4. **Calculating the Dot Product**:
Now, calculate the dot product:
\[ (x \hat{i}) \cdot (2 \hat{i}) + (\hat{j}) \cdot (3 \hat{j}) + (-\hat{k}) \cdot (4 \hat{k}) = 0 \]
This simplifies to:
\[ 2x + 3 - 4 = 0 \]

5. **Solving for \( x \)**:
Rearranging the equation gives:
\[ 2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]

### Final Answer:
The value of \( x \) is \( \frac{1}{2} \).

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