Magnetic force

Moving charges (i.e., currents) will exert a magnetic force on each other. This is given by Ampère’s force law:

$${\vec{F}}_{12}=i_{2}d{\vec{l}}_2\times \left(\frac{{\mu }_0}{4\pi }\frac{i_{1}d{\vec{l}}_2\times ({\vec{r}}_2-{\vec{r}}_1)}{\Vert {\vec{r}}_2-{\vec{r}}_1{\Vert }^3}\right)=\int i_2(d{\vec{l}}_2\times {\vec{B}}_{12})$$

where the term inside the brackets is the Biot-Savart law, and where $i_1,i_2$ are current-carrying wires.

A charged particle (with charge $q$ and velocity $\vec{v}$) moving through a magnetic field also has a magnetic force act on it. This is given by the Lorentz force law:

$${\vec{F}}_B=q(\vec{v}\times \vec{B})$$

where $\vec{B}$ is the magnetic flux density.

Observations

Note that the Lorentz force law indicates magnetic forces do no work, because the force is perpendicular to the motion of the particle. For a charge that moves $d\vec{l}=\vec{v}dt$:

$$dW=\vec{F}\cdot d\vec{l}=q(\vec{v}\times \vec{B})\cdot \vec{v}dt=0$$

The force also functionally acts as a centripetal force:

$${\vec{F}}_n=qvB\hat{n}=\frac{m{v}^2}{r}\hat{n}$$

i.e., we can link the radius of motion (which will be circular) with the mass of the charged particle. All of this is pretty helpful in describing the behaviour of DC motors and cyclotrons.

In general, “electric forces are enormously larger than magnetic ones”.¹ This is due to the sizes of the fundamental constants in the equations.

Derivation

For the Lorentz force law, say we have a magnetic field $\vec{B}$ and a charged particle with charge $q$ and velocity $\vec{v}$. From Ampère’s force law, we say that the particle’s displacement after a differential time $t$ is $d\vec{l}=\vec{v}dt$.

For this segment $d\vec{l}$, there is a current $i=\frac{q}{dt}$. Then:

$$id\vec{l}=q\frac{d\vec{l}}{dt}=q\vec{v}$$

Then we get the Lorentz force law, as stated.

See also

• Magnetism

Footnotes

1. From Introduction to Electrodynamics, by David J. Griffiths. ↩