Question

The charged particle moving in a uniform magnetic field of \((3\hat{i} + 2\hat{j}) \text{ T}\) has an acceleration \((4\hat{i} - \frac{x}{2}\hat{j}) \text{ m/s}^2\). The value of \(x\) is ____.

Answer 1

Correct Answer: 12

Step 1: Understanding the Question:
A magnetic force \(\vec{F} = q(\vec{v} \times \vec{B})\) is always perpendicular to both the velocity (\(\vec{v}\)) and the magnetic field (\(\vec{B}\)). Since acceleration (\(\vec{a}\)) is in the direction of the force, \(\vec{a}\) must also be perpendicular to \(\vec{B}\).
Step 2: Key Formula or Approach:
If two vectors are perpendicular, their dot product is zero:
\[ \vec{a} \cdot \vec{B} = 0 \]
Step 3: Detailed Explanation:
Given:
\(\vec{B} = 3\hat{i} + 2\hat{j}\)
\(\vec{a} = 4\hat{i} - \frac{x}{2}\hat{j}\)
Perform the dot product:
\[ (4\hat{i} - \frac{x}{2}\hat{j}) \cdot (3\hat{i} + 2\hat{j}) = 0 \]
\[ (4 \times 3) + \left(-\frac{x}{2} \times 2\right) = 0 \]
\[ 12 - x = 0 \]
\[ x = 12 \]
Step 4: Final Answer:
The value of \(x\) is \(12\).