Effects of the Lorentz Force

A very typical effect of a magnetic field is to make charged particles move in a circle. This is because the magnetic force is always perpendicular to the velocity of the particle, exactly what is required to generate circular motion.

For a particle moving with speed $v$ in a uniform magnetic field of strength $B$, equating the centripetal force for circular motion and the Lorentz force gives:

\[\frac{mv^2}{r} = \vert q\vert vB,\]

where $r$ is the radius of the circle. This gives the radius of the circle as:

\[r = \frac{mv}{\vert q \vert B}.\]

This radius is sometimes known as the gyroradius, or the Larmor radius.

This ability to make charged particles move in a circle is used in many applications, including:

  • Cyclotrons and synchrotrons to accelerate particles to high energies for research and medical purposes.
  • Mass spectrometers to separate ions by mass.

Magnetic fields from steady currents

Let’s imagine a steadily flowing current through a long wire, as already discussed in lecture 6. What magnetic field is generated?

We start from the result in the previous lecture, that for each individual charge $q$ moving at velocity $\vec{v}$, the magnetic field is given by:

\[\vec{B} = \frac{\mu_0}{4\pi} \frac{q}{\vert \vec{r} \vert^2} \vec{v} \times \hat{\vec{r}}\,.\]

In the lecture we will transform this equation to the case of a current $I$ flowing through a wire, using the following steps:

  • work out the ${\vec{B}}$ field by superposing the effect of $N$ charges with a mean drift velocity $\vec{v}_d$ (see lecture 6 for the idea of the drift velocity);
  • consider a segment of the wire with length $dL$ and cross-sectional area $A$;
  • note that the drift velocity must be along the direction of the wire;
  • use the relationship to the current $I = n_e e A v_d$, where $n_e$ is the number density of free electrons, and $e$ is the magnitude of the electron charge;
  • integrate over the wire to get the total magnetic field.

The result is that the magnetic field generated by a current $I$ flowing through a wire is given by the Biot-Savart law:

\[\vec{B} = \frac{\mu_0 I }{4\pi} \int_L \frac{d\vec{L} \times \hat{\vec{r}}}{\vert \vec{r} \vert^2}\,.\]

Here, $d\vec{L}$ is a small segment of the wire, with the vector pointing along the direction of the wire, and the integral goes along the whole wire.

What is the difference between this expression of the Biot-Savart law and the one for a moving charge? Two things:

  • We replace $q \vec{v}$ with $I {\rm d}\vec{L}$, where ${\rm d}\vec{L}$ is the length and direction of a single small segment of the wire;
  • We integrate over the whole wire to get the total magnetic field.

Discussion point: how do we know the drift velocity is along the wire? What would happen if it were not?

Examples starting from the Biot-Savart law

If the wire is completely straight and infinitely long1, say along the positive $z$-axis, the magnetic field circles around the wire in the $x$-$y$ plane. We will calculate the magnetic field at a distance $R$ from the wire in the lecture, showing that it is:

\[\vec{B} = \frac{\mu_0 I}{2\pi R} \hat{\vec{\phi}}\,,\]

where $\hat{\vec{\phi}}$ is a unit vector pointing anticlockwise in a circle around the wire in the $x$-$y$ plane, according to the right-hand rule.

But there is no need for a current to be straight; the wire it is flowing through can easily be curved. We can still apply the Biot-Savart law by adding up the effect of lots of small segments of wire; the individual segments locally look straight. For an example of this see e.g. Y&F section 28.5, where it is shown that the magnetic field due to a loop of wire in the $x=0$ plane, centred on $y=z=0$, is

\[\vec{B} = \frac{\mu_0 I a^2}{2(x^2 + a^2)^{3/2}} \hat{\vec{i}}\]

where $a$ is the radius of the loop, and the field is measured at $\vec{r}=(x,0,0)$.

Note for $x=0$ (in the very middle of the loop) this simplifies to

\[\vec{B} = \frac{\mu_0 I}{2 a} \hat{\vec{i}}\]

Superposition

Because of superposition, magnetic fields from multiple currents add up and can produce complicated magnetic field patterns. Below, you can play with the magnetic field before and after superposition to understand how this works.

With parallel currents (the $\odot$ now indicates currents in a straight wire, coming out of the page):

With anti-parallel currents (the $\otimes$ now indicates currents in a straight wire, going into the page):

This gives us a way to understand the effect of a loop of current too. Imagine there is a current coming out the page, looping round in front, going back into the page, and then round the back. The magnetic field you get from superposing the outcoming and incoming currents give the main contribution to the magnetic field in the plane, so we get roughly the right magnetic field pattern. Here it is again with fieldlines instead of arrows:

This looks very much like the magnetic field from a bar magnet – not a coincidence, since a bar magnet acts basically like a loop of current, as we will see.

Forces on current-carrying wires

Just as we can average over the magnetic field from lots of moving charges to work out the field from a current, we can sum over the forces on all these charges to work out the force on the wire.

  • Since the problem consists of an infinite number of little segments of wire again, the ‘sum’ really turns into an integral
  • The starting point is now $\vec{F} = q \vec{v} \times \vec{B}$
  • Making the same transformations as we did for the Biot-Savart law, we can get the net force in a little segment ${\rm d}\vec{L}$ in terms first of the electron density etc, then substituting to make it purely in terms of the current.
  • On doing this, you should find ${\rm d}\vec{F} = I {\rm d}\vec{L} \times \vec{B}$.
  • Then, you integrate to get the net force on the wire, $\vec{F} = I \int_L {\rm d} \vec{L} \times \vec{B}$.

If stuck, see YF section 27.6.

In summary, the generalization from single charges to currents is as follows:

  Single moving charge Current
Field generated $\vec{B} = \frac{\mu_0}{4\pi} \frac{q}{\vert \vec{r} \vert^2} \vec{v} \times \hat{\vec{r}}$ $\vec{B} = \frac{\mu_0 I }{4\pi} \int_L \frac{d\vec{L} \times \hat{\vec{r}}}{\vert \vec{r} \vert^2}$
Force experienced $\vec{F} = q \vec{v} \times \vec{B}$ $\vec{F} = I \int_L {\rm d} \vec{L} \times \vec{B}$

Footnotes

  1. Of course nothing is really infinitely long. But often we talk about infinitely long wires, meaning that the wire is much longer than any distances we are interested in. This is a common approximation in physics. ↩