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# Movement of a Charge in a Magnetic Field

An analysis of the movement of an electric charge in a magnetic field, also discussing the particular case when the charge travels at right angles relative to the field
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Contents

### Key Facts

Gyroscopic Couple: The rate of change of angular momentum () = (In the limit).
• = Moment of Inertia.
• = Angular velocity
• = Angular velocity of precession.

## Overview

Key facts

The force acting on a charge traveling at a speed in a magnetic field is given by:

If a charged particle of mass travels with a speed at right angles relative to a magnetic field , then the charge will move in a circle of radius given by:

In order to define the force acting on an electric charge placed in a magnetic field, consider a charge traveling at a speed in a magnetic field . Also, assume that in a time the charge traveled the distance , as highlighted in Figure 1.

We can thus write that:

from which:

Also, we know that the electric current can be calculated as:

where is the electric charge transfered during a time . Using the expression of time from (3), equation (4) becomes:

The force which acts on an electric charge responsible for generating a current , and which is placed in a magnetic field , can be written as:

By using the expression form of the electric current from (5) in (6), we get that: