Question

A particle of mass \( m \) and charge \( q \) is moving with velocity \( \mathbf{v} = v_x \hat{i} + v_y \hat{j} \). If it is subjected to a magnetic field \( \mathbf{B} = B_0 \hat{i} \), it will move in a:

• A. straight line path
• B. circular path
• C. helical path
• D. parabolic path

Hint:

A charged particle moves in a helical path when its velocity has a component parallel and perpendicular to the magnetic field.

Answer 1

The Correct Option is C

Motion of a Charged Particle in a Magnetic Field
- The Lorentz force acting on a charged particle within a magnetic field is defined as:\[\mathbf{F} = q (\mathbf{v} \times \mathbf{B})\]- With \( \mathbf{B} = B_0 \hat{i} \) and \( \mathbf{v} = v_x \hat{i} + v_y \hat{j} \), the cross-product is computed as:\[\mathbf{F} = q [(v_x \hat{i} + v_y \hat{j}) \times (B_0 \hat{i})]\]- Utilizing the properties \( \hat{i} \times \hat{i} = 0 \) and \( \hat{j} \times \hat{i} = -\hat{k} \), the force simplifies to:\[\mathbf{F} = -q v_y B_0 \hat{k}\]- This force is perpendicular to the \( y \)-component of velocity, inducing circular motion in the \( yz \)-plane.- The \( x \)-component of velocity is unaffected, resulting in a helical trajectory.Consequently, the appropriate conclusion is (C) helical path.