A charged particle is moving in a uniform magnetic field. if the direction of motion of the charged particle…

1. Analyze the question

The question asks for the trajectory (path) of a charged particle when it moves through a uniform magnetic field, specifically when its velocity vector is parallel to the magnetic field lines.

2. Explain the concept

The force ($F$) exerted on a charged particle moving in a magnetic field is given by the Lorentz force formula:

Where:

• $q$ = charge of the particle
• $v$ = velocity of the particle
• $B$ = magnetic field strength
• $\theta$ = the angle between the velocity vector ($v$) and the magnetic field vector ($B$)

When the motion is parallel to the magnetic field, the angle $\theta = 0^\circ$. Since $\sin(0^\circ) = 0$, the force $F$ becomes:

Because there is no magnetic force acting on the particle, it experiences no acceleration or change in direction. According to Newton’s First Law of Motion, an object in motion will continue in a straight line unless acted upon by an external force.

3. Show the solution

• Step 1: Identify the angle between the velocity and the magnetic field. Since they are parallel, $\theta = 0^\circ$.
• Step 2: Apply the magnetic force formula $F = qvB \sin(\theta)$.
• Step 3: Calculate the force: $F = qvB \sin(0^\circ) = 0$.
• Step 4: Conclude the effect on motion. With zero net force, the particle maintains its constant velocity and direction. Therefore, it moves in a straight line.

4. Identify the answer

The correct option is A.

5. Exam Tip

Remember the "Sine Rule" for magnetic force:

• If $\theta = 0^\circ$ or $180^\circ$ (parallel/anti-parallel), $\sin(\theta) = 0$, so no force (straight line).
• If $\theta = 90^\circ$ (perpendicular), $\sin(\theta) = 1$, so maximum force (circular path).
• If the angle is between $0^\circ$ and $90^\circ$, the path is a helix (a spiral).