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 that the magnetic field \( \mathbf{B} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The force \( \mathbf{F} \) on a charged particle moving in a magnetic field is given by the equation:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the particle, \( \mathbf{v} \) is its velocity, and \( \mathbf{B} \) is the magnetic field. The force \( \mathbf{F} \) is always perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \). Consequently, the acceleration \( \mathbf{a} \) is also perpendicular to \( \mathbf{B} \).

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

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

4. **Calculating the Dot Product**:
Calculate the dot product:
\[ = x \cdot 2 + 1 \cdot 3 + (-1) \cdot 4 \]
This simplifies to:
\[ 2x + 3 - 4 = 0 \]
Therefore:
\[ 2x - 1 = 0 \]

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

### Final Answer:
The value of \( x \) is \( 0.5 \).

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