Understanding Charge Movement in a Uniform Magnetic Field

Let's analyze the path of a charge moving parallel to a uniform magnetic field. This involves understanding the magnetic force that acts on a moving charge.

The force experienced by a charge ($q$) moving with velocity ($\vec{v}$) in a magnetic field ($\vec{B}$) is given by the Lorentz force formula for the magnetic component:

\begin{equation*} \vec{F}_B = q(\vec{v} \times \vec{B}) \end{equation*}

The magnitude of this magnetic force is given by:

\begin{equation*} F_B = |q| v B \sin(\theta) \end{equation*}

where:

• $|q|$ is the magnitude of the charge
• $v$ is the magnitude of the velocity of the charge
• $B$ is the magnitude of the magnetic field
• $\theta$ is the angle between the velocity vector ($\vec{v}$) and the magnetic field vector ($\vec{B}$)

In this question, the charge is moving parallel to a uniform magnetic field. This means the velocity vector $\vec{v}$ is parallel to the magnetic field vector $\vec{B}$. The angle $\theta$ between $\vec{v}$ and $\vec{B}$ can be either $0^{\circ}$ (if moving in the same direction as the field) or $180^{\circ}$ (if moving in the opposite direction to the field).

Let's calculate the magnetic force ($F_B$) for these angles:

• If $\theta = 0^{\circ}$, $\sin(0^{\circ}) = 0$. So, $F_B = |q| v B (0) = 0$.
• If $\theta = 180^{\circ}$, $\sin(180^{\circ}) = 0$. So, $F_B = |q| v B (0) = 0$.

In both cases where the charge moves parallel (or anti-parallel) to the uniform magnetic field, the magnetic force acting on the charge is zero ($F_B = 0$).

According to Newton's first law of motion, if there is no net force acting on an object (the charge, in this case), its velocity remains constant. This means if it was moving, it will continue to move with the same speed in the same direction. Therefore, its path will be a straight line.

Let's briefly consider why the other options are incorrect paths for a charge moving parallel to a uniform magnetic field:

• Circular path: A circular path occurs when the velocity is perpendicular ($\theta = 90^{\circ}$) to a uniform magnetic field. The magnetic force acts as a centripetal force.
• Helical path: A helical path occurs when the velocity has a component parallel to the magnetic field and a component perpendicular to it ($0^{\circ} < \theta < 90^{\circ}$ or $90^{\circ} < \theta < 180^{\circ}$). The parallel component causes linear motion, while the perpendicular component causes circular motion around the field line, resulting in a helix.
• Elliptical path: An elliptical path is not typically generated by a uniform magnetic field alone acting on a charge.

Since the magnetic force is zero when the charge moves parallel to the uniform magnetic field, there is no force to change the direction of its velocity. Hence, the path remains a straight line.

Revision Table: Charge Path in Uniform Magnetic Field

Additional Information: Lorentz Force and Charge Motion

The Lorentz force is the total force on a charged particle due to electromagnetic fields. It has two components: an electric force and a magnetic force.

\begin{equation*} \vec{F} = q\vec{E} + q(\vec{v} \times \vec{B}) \end{equation*}

where:

• $\vec{F}$ is the total force
• $q$ is the charge of the particle
• $\vec{E}$ is the electric field vector
• $\vec{v}$ is the velocity vector of the particle
• $\vec{B}$ is the magnetic field vector

In the context of this question, we only considered the magnetic force component ($q(\vec{v} \times \vec{B})$) because only a magnetic field is mentioned, and we assumed no electric field ($\vec{E} = 0$). The resulting path of a charge in a magnetic field is solely determined by the magnetic force, which always acts perpendicular to both the velocity of the charge and the magnetic field direction.

When a charge moves parallel to the magnetic field, the cross product $\vec{v} \times \vec{B}$ is zero, resulting in zero magnetic force. Thus, the particle experiences no deflection and continues its motion in a straight line, conserving its momentum in that direction.