# Does a charged particle travelling with uniform velocity induce a magnetic field?

Does a charged particle, an electron say, travelling with uniform velocity induce a magnetic field? I believe it doesn't. In primary school, we all learned how to induce a magnetic field into an iron nail by wrapping coils of wire around the nail and then hooking it up to a DC battery, but if you do not coil the wire, the magnetic nail doesn't occur. What's happening here? My only guess are the electrons are accelerating; the magnitudes of their speeds aren't changing, but rather their directions. In the coil, a force must be applying itself to the electrons in order for them to make their spiralling paths, thus, they are said to be accelerating and that is what causes the magnetic field to develop.

• Running a current down a straight wire does create a magnetic field. – NickD Apr 26 '18 at 0:16
• @Nick Excellent, please tell me why. – Michael Lee Apr 26 '18 at 0:34
• I'm happy to know I'm wrong and happy to know why. – Michael Lee Apr 26 '18 at 0:34
• Why is b/c Maxwell's equations. Have considered looking at the Lorentz boosted electrostatic field of a point charge? – JEB Apr 26 '18 at 0:56
• @JEB Thank you so much, you've given me much reading material on the subject matter! – Michael Lee Apr 26 '18 at 1:00

A straight wire does have a magnetic field. It circles around the wire instead of going in a straight line like in a coil.

On the left is a straight wire with the magnetic field curling around it. The middle shows a single loop of wire. Notice that the magnetic field still curls around the wire, but the fields from opposite ends of the loop add together to make a strong field. The right picture shows a multi-loop wire (a solenoid), which enhances the field compared to the single loop. The right picture is the kind of field you created with the wire and nail. For the same current, the solenoid creates a much stronger field, which is why it is used to magnetize the nail.

To answer your original question, a single electron in motion does have a magnetic field that's similar to the straight wire (the field curls around the electron's path of motion) except that it gets weaker as you move farther away along the electon's path.

• How does this interact with relativity? In the particle's reference frame it's not moving, so there should be no magnetic field (IIRC). Are magnetic fields dependent on reference frame? – Kevin Apr 26 '18 at 18:17
• @Kevin Yes. In different reference frames, magnetic fields can look like electric fields and vice versa. One of the consequences of relativity is that these two fields are unified into a single electromagnetic field. – Mark H Apr 26 '18 at 20:02

A flow of electrons makes a current and therefore produces a magnetic field.

If electrons are traveling with a uniform speed, the current they form produces a constant magnetic field.

The reason you need a coil to magnetize a nail is because the magnetic fields of individual loops in a coil add up on the inside of the coil and create a strong unidirectional magnetic field there, with straight magnetic field lines. So, when a nail is inserted in such coil, it is permeated with the dense magnetic field all along its length.

In comparison, the magnetic field around a strait current carrying wire of the same length that was used to make the coil, is spread out over the length of the wire and, as a result, the density of the magnetic field is relatively low. In addition, magnetic filed lines around a straight wire are circular. So, there is no place around the wire where a nail could be exposed to a strong magnetic field aligned with its body.

Nevertheless, such field certainly exists, which can be easily demonstrated by using a compass or many other methods.

My only guess are the electrons are accelerating; the magnitudes of their speeds aren't changing, but rather their directions.

Wrong guess. Here is the definition of the magnetic field,exactly based on the force induced on a moving test charge

Experiments show that the magnetic field of a moving charge can be expressed as:

$μ_0$ is called the permeability of free space. The constant $ε_o$ that is used in electric field calculations is called the permittivity of free space. Note that $ε_oμ_o$ = 1/c2.

Just like the charge of the electron is an experimental fact, the magnetic field around a moving charge has been measured by experiment.

You state:

In the coil, a force must be applying itself to the electrons in order for them to make their spiralling paths.

The force comes from the electric fields in the solid state lattice of the coil, with which individual electrons in the current interact.

thus, they are said to be accelerating and that is what causes the magnetic field to develop.

Acceleration of electrons induces radiation, not a magnetic field, whether linear or circular. In a current carrying wire it is not one electron running around in the coil, but many drifting along the lattice of the wire, building up the current in the coil incrementally. The magnetic field of individual electrons is following this incremental steps, according to the formula above and the integration builds the magnetic field of the coil.

• The velocity dependence would mean magnetic fields are dependent on reference frame then, right? How does that work? Are there any other fields that happens with? – Kevin Apr 26 '18 at 18:25
• Is that why old incandesce light bulbs consist of fine coils of wire? What if we took two electrons and placed them very close together, in an otherwise empty universe, and suddenly release them like releasing a pulled rubber band. What would happen? – Michael Lee Apr 29 '18 at 21:04
• electrons are quantum mechanical, they would repel according (numbers) to the diagram hyperphysics.phy-astr.gsu.edu/hbase/Particles/imgpar/feynm2.gif , and when the distance is macroscopic could be modeled as a classical point particle. – anna v Apr 30 '18 at 4:22
• @MichaelLee the thin wires are for the increase in resistance and the subsequent heating from the drifting electrons in the current hitting more atoms due to the smaller crossection of the wire. The current is carried on by the same number of drifting electrons . – anna v Apr 30 '18 at 4:32
• In regards to Kevin's comment above, I had a professor that explained that magnetism is a "relativistic effect" of moving charges (electricity). If anyone has more "intuitive" insight into this it would be helpful. – Jack R. Woods Apr 30 '18 at 14:22

Of course there is a magnetic field, otherwise the ampere would not exist. The ampere is defined by the magnetic attraction of two parallel straight wires carrying DC current.

(This answer is backwards in that it explains the existence of the field in terms of the non-silliness of the SI.)

• Note that after the upcoming SI redefinition the ampere will be defined in terms of fundamental charges, like most people probably think it already is. – rob Apr 30 '18 at 6:41