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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
Contents

Overview

Key facts

The force acting on a charge $\inline&space;q$ traveling at a speed $\inline&space;v$ in a magnetic field $\inline&space;B$ is given by:

$F&space;=&space;q&space;v&space;B$

If a charged particle of mass $\inline&space;m$ travels with a speed $\inline&space;v$ at right angles relative to a magnetic field $\inline&space;B$, then the charge will move in a circle of radius $\inline&space;r$ given by:

$r&space;=&space;\frac{mv}{qB}$

In order to define the force acting on an electric charge placed in a magnetic field, consider a charge $\inline&space;q$ traveling at a speed $\inline&space;v$ in a magnetic field $\inline&space;B$. Also, assume that in a time $\inline&space;dt$ the charge traveled the distance $\inline&space;dl$, as highlighted in Figure 1.

We can thus write that:

$v&space;=&space;\frac{dl}{dt}$

from which:

$dt&space;=&space;\frac{dl}{v}$

Also, we know that the electric current $\inline&space;i$ can be calculated as:

$i&space;=&space;\frac{q}{t}$

where $\inline&space;q$ is the electric charge transfered during a time $\inline&space;t$. Using the expression of time from (2), equation (3) becomes:

$i&space;=&space;\frac{qv}{dl}$

The force which acts on an electric charge $\inline&space;q$ responsible for generating a current $\inline&space;i$, and which is placed in a magnetic field $\inline&space;B$, can be written as:

$F&space;=&space;i&space;\cdot&space;dl&space;\cdot&space;B$

By using the expression form of the electric current from (4) in (5), we get that:

$F&space;=&space;\frac{qv}{dl}&space;\cdot&space;dl&space;\cdot&space;B$

$F&space;=&space;q&space;v&space;B$

This force acts in the same direction as that of the lines of force of the magnetic field $\inline&space;B$. If the charge travels at right angles relative to the magnetic field, as for example in Figure 1, then the force will act perpendicular to the direction of the charge's movement, and thus the charge will move in a circle (see Figure 2).

As in this case the force $\inline&space;F=qvB$ is oriented inwards, it equivalates the centripetal force $\inline&space;F_c$ given by:

$F_c&space;=&space;\frac{mv^2}{r}$

where $\inline&space;m$ is the mass of the charged particle, and $\inline&space;r$ is the radius of the circle. Thus, we get that:

$\frac{mv^2}{r}&space;=&space;qvB$

from which we obtain the radius of the circle on which the charge moves:

$r&space;=&space;\frac{mv}{qB}$