Question

A charged particle is moving in a magnetic field \[ \vec B = (3\hat i + 2\hat j)\ \text{T} \] with acceleration \[ \vec a = 4\hat i - \frac{x}{2}\hat j\ \text{m/s}^2 \] Find \(x\).

Answer 1

Correct Answer: 12

Concept: Magnetic force on a charged particle is \[ \vec F = q(\vec v \times \vec B) \] Thus magnetic force is always perpendicular to the magnetic field. Since acceleration is along the direction of force, \[ \vec a \perp \vec B \] Hence \[ \vec a \cdot \vec B = 0 \] Step 1: Compute dot product \[ (4\hat i - \frac{x}{2}\hat j)\cdot(3\hat i + 2\hat j) = 0 \] \[ 12 - x = 0 \] \[ x = 12 \]