Question

The magnetic force \( q [v \times B] \) is

• A. parallel to both \( v \) and \( B \)
• B. perpendicular to \( v \)
• C. perpendicular to both \( v \) and \( B \)
• D. parallel to \( B \)

Hint:

The direction of the magnetic force can be found using the right-hand rule. Point your fingers in the direction of \( \vec{v} \), curl them towards \( \vec{B} \), and your thumb will point in the direction of \( \vec{F} \), which is perpendicular to both \( \vec{v} \) and \( \vec{B} \).

Answer 1

The Correct Option is C

The magnetic force on a charged particle moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by: \[ \vec{F} = q (\vec{v} \times \vec{B}) \] Where: - \( q \) is the charge of the particle, - \( \vec{v} \) is the velocity vector, - \( \vec{B} \) is the magnetic field vector. This force is always perpendicular to both the velocity \( \vec{v} \) and the magnetic field \( \vec{B} \). Thus, the force is perpendicular to both \( \vec{v} \) and \( \vec{B} \), which is the correct option.

Answer 2

Step 1: Understanding the magnetic force formula
The magnetic force on a charged particle moving in a magnetic field is given by the vector cross product \( \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \), where:
- \( q \) is the charge of the particle,
- \( \mathbf{v} \) is the velocity vector of the particle,
- \( \mathbf{B} \) is the magnetic field vector.

Step 2: Properties of the cross product
The cross product \( \mathbf{v} \times \mathbf{B} \) produces a vector that is perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \).

Step 3: Direction of magnetic force
Therefore, the magnetic force acts in a direction perpendicular to both the velocity of the charged particle and the magnetic field.

Step 4: Conclusion
Hence, the magnetic force \( q [\mathbf{v} \times \mathbf{B}] \) is perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \).