A charged particle has acceleration `veca=2hati+xhatj` in a magnetic field `vecB=-3hati+2hatj-4hatk`.Find the value of `x`.

Options

A. 4
B. 2
C. 6
D. 3

Answer

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

### Step-by-Step Solution:

1. **Understand the relationship between acceleration and magnetic field**:
When a charged particle moves in a magnetic field, the force acting on it is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
where \( q \) is the charge of the particle, \( \vec{v} \) is its velocity, and \( \vec{B} \) is the magnetic field. The acceleration \( \vec{a} \) of the particle is related to the force by Newton's second law:
\[ \vec{F} = m \vec{a} \]
Therefore, we can conclude that the acceleration \( \vec{a} \) is perpendicular to the magnetic field \( \vec{B} \).

2. **Set up the dot product**:
Since the acceleration \( \vec{a} \) is perpendicular to the magnetic field \( \vec{B} \), their dot product must equal zero:
\[ \vec{B} \cdot \vec{a} = 0 \]

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

4. **Calculate the dot product**:
The dot product can be calculated as follows:
\[ (-3)(2) + (2)(x) + (-4)(0) = 0 \]
Simplifying this gives:
\[ -6 + 2x = 0 \]

5. **Solve for \( x \)**:
Rearranging the equation to solve for \( x \):
\[ 2x = 6 \implies x = \frac{6}{2} = 3 \]

Thus, the value of \( x \) is \( 3 \).

### Final Answer:
\[ x = 3 \]