Chapter 26.1, Problem 4PP

To determine

The trajectory that a positive ion would follow moving in a magnetic field.

Answer to Problem 4PP

The ion will follow helical path.

Explanation of Solution

Introduction:

Magnetic field is a region which defines the magnetic effect of electrical charges and magnetized materials in relative motion. When a charged particle or a conductive current is placed in a magnetic field, it experiences a force.

When a charged particle moves in a magnetic field, the magnetic force on a charged particle $q$ depends on the angle $\theta$ where $\theta$is the angle between the velocity vector, $\overrightarrow{v}$ and the magnetic field vector $\overrightarrow{B}$ .

The magnetic force is given by:

  $\left|\overrightarrow{F}\right|=qvB\sin \theta$........................... (1)

Here, $F,\theta ,v$ and $B$ are force, angle between direction of velocity and direction of magnetic field.

If the charge is moving parallel to the field $B$ , then $\theta =0°$ .

Substitute $0°$ for $\theta$ in Eq. (1)

  $\begin{array}{l}\left|\overrightarrow{F}\right|=qvB\sin 0°\\ \boxed{\left|\overrightarrow{F}\right|=0}\end{array}$

Therefore, the force experienced by the charged particle will be zero and the motion of the particle will remain non-deflected.

When charged particles moving perpendicular to magnetic fields, then the angle between the direction of velocity and magnetic field is $90°$ .

Substitute $90°$ for $\theta$ in Eq. (1)

  $\begin{array}{l}\left|\overrightarrow{F}\right|=qvB\sin 90°\\ \boxed{\left|\overrightarrow{F}\right|=qvB}\end{array}$

In this case, trajectory will be circular.

Again,

If the magnetic field varies with time then the positive charge vector will become a resulting vector of two velocity components. In this case the ion’s trajectory will be helical.

Conclusion:

Hence, the trajectory of the ion in a magnetic where magnetic field is changing linearly with time will be helical path.