Question

What will be the path followed by a charged particle moving along a magnetic field.

Hint:

Remember the conditions for different paths in a uniform magnetic field: - \(\theta = 0^\circ\) or \(180^\circ\) (parallel/anti-parallel): \textbf{Straight Line} (zero force). - \(\theta = 90^\circ\) (perpendicular): \textbf{Circular Path} (maximum force). - Any other angle \(0<\theta<180\), \(\theta \neq 90^\circ\): \textbf{Helical Path} (a spiral).

Answer 1

Step 1: Understanding the Concept:
A charged particle moving in a magnetic field experiences a magnetic force (Lorentz force) only if its velocity has a component perpendicular to the direction of the magnetic field. The magnitude and direction of this force determine the particle's trajectory.
Step 2: Key Formula or Approach:
The magnetic force \(\vec{F}\) on a particle with charge \(q\) moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) is given by the vector cross product:
\[ \vec{F} = q(\vec{v} \times \vec{B}) \] The magnitude of this force is given by:
\[ F = |q|vB\sin(\theta) \] where \(\theta\) is the angle between the velocity vector \(\vec{v}\) and the magnetic field vector \(\vec{B}\).
Step 3: Detailed Explanation:
In this specific case, the charged particle is "moving along the magnetic field." This means its velocity vector \(\vec{v}\) is parallel to the magnetic field vector \(\vec{B}\).
When two vectors are parallel, the angle \(\theta\) between them is 0\(^{\circ}\).
Let's calculate the magnitude of the force using this angle:
\[ F = |q|vB\sin(0^\circ) \] Since the value of \(\sin(0^\circ)\) is 0:
\[ F = |q|vB(0) = 0 \] If the particle were moving anti-parallel to the field, the angle would be 180\(^{\circ}\), and \(\sin(180^\circ)\) is also 0, resulting in zero force.
Since the net magnetic force acting on the particle is zero, there is no change in its state of motion according to Newton's first law. The particle will continue to move with its constant velocity, which means it will follow a straight-line path.
Step 4: Final Answer:
Because the magnetic force on the particle is zero when it moves parallel to the magnetic field, its path will be an undeflected straight line.