Why does a current-carrying conductor behave like a magnet?

I have heard that a current-carrying conductor behaves like a magnet, the reason why the magnetic needle of a compass kept near the circuit, deflects. Why is it so?

Moving charges generate a magnetic field, and the current in the conductor consists of moving charges.

• This might be better if you included something about Ampere's law (in order to explain why moving charges generate magnetic fields). Jul 30, 2015 at 16:29

A current passing through a wire generates a magnetic field. This is explained by special relativity. Similar to the combination of space and time into the four-component spacetime vector magnetic and electric field are combined into the electromagnetic tensor. Changing reference frames mixes the components between electric and magnetic field.

You can say one electron in its reference frame just observes his electric field, but a passing, second electron sees this partially as a magnetic field.

Electrons and protons have permanently an electric field. To produce a macroscopic electric field it is enough to separate charged particles. But what is the source of a magnetic field? There are two ways to produce an magnetic field. Both ways based on the second phenomenon of charged particles. Electrons, protons and other particles too have magnetic dipole moments and parallel to this moment an intrinsic spin.

The first way to produce magnetic field is to bend a current carrying wire. The intrinsic spin of the flowing electrons due to the gyroscopic effect align the electrons and by this the magnetic dipole moments align too. Now instead of equally distributed magnetic dipole moments the electron's magnetic dipole moments are aligned and it produces a macroscopic magnetic dipole. The second way is the influence of an external magnetic field. Electrons get aligned under its influence.

In a straight current carrying wire we have some side effects. First, to prove the existence of a magnetic field one usual use a magnet (compass needle). This induces a magnetic field due to the alignment of the electrons and more due to Lorentz force and folowing selfinductance. Second, the straight wire is not straight "behind the horizon". On the way from the source to the sink the wire is bended anyway and this led to a magnetic field too.

• It would also be fruitful if you explain in what manner do the electrons/ charged particles get aligned. Aug 9, 2015 at 8:28
• &Sashank Sriram "The intrinsic spin of the flowing electrons due to the gyroscopic effect align the electrons and by this the magnetic dipole moments align too." is the key to understand it. Aug 9, 2015 at 9:15

A current flow in a conductor is due to a difference in the electric potential between the two ends of the circuit. All charges subject to an electric field experience an electric force, which according to Newton's law means that they will be accelerated (even though the drift velocity is approximately constant). Accelerated charges generate a magnetic field, which can be understood via Maxwell's equations. Considering how your question was posed I guess you don't have advanced electrodynamics knowledge, but if you look at the equation governing the curl of the magnetic field

$$\nabla \times \vec{H} = \frac{4\pi}{c}\vec{J} + \frac{1}{c}\frac{\partial \vec{D}}{\partial t}$$

where $\vec{H}$ is the magnetic field and $\vec{J}$ is the current flowing in the wire and $\vec{D}$ is the electric displacement.

You can consider a current as composed of the individual electric charges circulating, which means that the term

$$\frac{1}{c}\frac{\partial \vec{D}}{\partial t}$$

is non zero due to the electric displacement varying with respect to a resting frame. Any moving charge will generate a magnetic field due to this displacement current.

A wire is neutral in the macroscopic scale, but locally it is not. So the term $\vec{J}$ is actually a macroscopic effective behavior due to local variations of the electric displacement.

• Yes, I was wondering about it when it was mentioned in my school textbook. Thank you for your explanation. Instead of going through both the Bio-Savart law and the Maxwell Equations pointed out by a gentleman here, yours was more understandable, with a little help in some places. Jul 30, 2015 at 16:13
• No problem. May I politely ask you to up-vote if the answer pleased you? :) Jul 30, 2015 at 16:22
• Why does this answer focus on the displacement instead of just $J$? Charges moving in a wire give you a nonzero $J$ which trivially cause a nonzero magnetic field. Aug 9, 2015 at 15:57
• If you take it to the last consequences current flowig in wires are point particles with a mean drift velocity, the term $J$ is non zero only at the particle when Ampere's law is at the differential form. Aug 9, 2015 at 16:16