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:
The topic is the Magnetic Force on a moving charge.
The magnetic force, given by \(\vec{F} = q(\vec{v} \times \vec{B})\), is always perpendicular to the magnetic field \(\vec{B}\). Since \(\vec{F} = m\vec{a}\), the acceleration vector \(\vec{a}\) must also be perpendicular to \(\vec{B}\).
Step 2: Key Formula or Approach:
The mathematical condition for two vectors to be perpendicular is that their dot product (scalar product) is zero.
Therefore, we must have:
\[ \vec{a} \cdot \vec{B} = 0 \]
Step 3: Detailed Explanation:
The given vectors are:
Magnetic field, \(\vec{B} = 3\hat{i} + 2\hat{j}\).
Acceleration, \(\vec{a} = 4\hat{i} - \frac{x}{2}\hat{j}\).
Now, we compute their dot product and set it to zero:
\[ \vec{a} \cdot \vec{B} = (4\hat{i} - \frac{x}{2}\hat{j}) \cdot (3\hat{i} + 2\hat{j}) = 0 \]
The dot product is calculated as \(a_x B_x + a_y B_y + a_z B_z\):
\[ (4)(3) + \left(-\frac{x}{2}\right)(2) = 0 \]
\[ 12 - x = 0 \]
Solving for \(x\), we get:
\[ x = 12 \]
Step 4: Final Answer:
The value of \(x\) is \(12\).