Electricity and magnetism/Magnetism

Magnets

Like electric forces, magnetic forces obey the principle of the attraction of opposites. Magnets always have two poles, a North Pole which is naturally attracted to the Earth's magnetic South Pole, and a South Pole, which is naturally attracted to the Earth's magnetic North Pole. In a compass, the North Pole of the magnetic needle indicates the geographic North Pole of the Earth. The geographic North and South are therefore reversed in relation to the magnetic North and South.

When magnets love each other, they do this in a somewhat special way. The North Pole of one sticks to the South Pole of the other, as if magnetism had chosen the eroticism of 69.

Unlike electric forces, isolated magnetic charges have never been found. Apparently Nature did not welcome magnetic monopoles. Magnets are always dipoles. If we cut a magnet in the middle between its two poles, we do not obtain two monopoles, but two new dipoles, two new magnets which each have two poles:

It is explained by assuming that a magnetic material is composed of microscopic magnets all aligned in the same direction:

Magnetostatics is the study of magnetic forces between stationary magnets. The calculation of magnetic forces is similar to that of electric forces, except that we reason on dipoles, and never on monopoles:

Like the electric field $\mathbf {E}$ for electric charges, the magnetic field $\mathbf {B}$ is like a mathematical intermediary that is used to calculate the forces between magnetized materials. It is also used to calculate the forces exerted by magnets on moving charges, the forces exerted by moving charges on magnets, and the forces exerted between moving charges. But it is also much more than a mathematical intermediary, because it has an autonomous existence.

Magnetic force is produced by electric currents

A magnet naturally orients itself perpendicular to an electric current. Its direction depends on the direction of the current. An electric current is therefore a source of magnetic force. This is a discovery made by Hans Christian Ørsted in 1820:

If we cut the current, this magnetic force disappears:

Ampère concluded that an electric current can behave like a magnet. This conclusion led him to the discovery of Ampère's law (1825):

Two parallel electrical wires carrying a current attract each other if the currents go in the same direction and repel each other if the currents go in the opposite direction.

The force between two electrical wires carrying a current is the magnetic force. It cannot be an electric force since the two wires are electrically neutral.

The unit of measurement of electric current, the ampere, was defined from the magnetic force between two wires carrying a current:

"The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10⁻⁷ newtons per metre of length." (International Bureau of Weights and Measures, 1948)

Since 1 A = 1 C/s, the definition of the ampere also defines the Coulomb.

When Ampère discovered his law, electrons were not known, so the ampere could not be defined as a current of electrons. Now that the electrons are known and their charge -q = -1.602 176 487 ×10⁻¹⁹ C has been precisely measured, we can define the ampere by the number of electrons it takes to make 1 C: 1 ampere = 6.241509074×10¹⁸ electrons per second, roughly six billion billion electrons every second.

Current loops behave like magnets. Two parallel loops attract each other if their currents go in the same direction and repel each other if they go in the opposite direction. A current loop is therefore a magnetic dipole:

The magnetic field produced by a current loop is similar to the electric field produced by an electric dipole:

Ampère assumed that the magnetic force of magnets is produced by microscopic current loops. We now know that it comes mainly from the spin of electrons. Electrons are magnetic dipoles because they behave like spinning tops. The rotation of electrons is not an electric current, but the effect is similar to that of a microscopic current loop.

As with the electric field, the magnetic field produced by several sources is the sum of the fields produced by each of them separately. For two parallel wires carrying currents in opposite directions, we obtain the total field by making a vector sum:

The Earth's magnetic field is produced at the center of the Earth by its constantly moving liquid iron core:

Coulomb's law in electrodynamics

To calculate the forces between moving charges we need generalized Coulomb's law:

If a charge is eternally stationary in an inertial frame of reference, then the force it exerts on a moving charge is the same as the electrostatic force it would exert on that charge if it were stationary at the same position.

If a charge is eternally immobile in an inertial frame of reference, the electric field of which it is the source has had time to propagate throughout space and establish itself there.

If a charge is accelerated, there is no inertial frame of reference in which it is eternally stationary. The Coulomb field of which it is the source in its rest frame at a given moment takes time to propagate. At a later time, it is the source of another Coulomb field, in another rest frame. This is why accelerated charges are the source of electromagnetic waves that propagate throughout space. Light is produced by accelerated electrical charges:

An immobile electric charge is not a source of light because no waves propagate in the electric field it establishes around it.

Generalized Coulomb's law makes it possible to calculate the electric force field of any system of charges in uniform rectilinear motion relative to each other. To calculate the force exerted by all the charges, we calculate the sum of the forces exerted separately by each charge. To calculate the force exerted by a charge, we consider the frame of reference where it is at rest and calculate the Coulomb force. The Lorentz transformation then makes it possible to calculate this force in any frame of reference.

(Animation: Coulomb field produced by a charge in uniform rectilinear motion)

The magnetic force of the electric current is a consequence of the Fitzgerald contraction and the Coulomb force

Fitzgerald contraction is the contraction of all solid bodies in the direction of their motion. It is appreciable only for bodies whose speed $v$ approaches the speed $c$ of light. The contraction factor is ${\sqrt {1-v^{2}/c^{2}}}$ .

When an electrically charged body is contracted, its charge density increases. This effect is at the origin of the magnetic force of the electric current. The relative movement of positive and negative charges can give rise to differences in charge density and therefore electrostatic forces. These electrostatic forces in one frame of reference are magnetic forces in another frame of reference.

We can make a model of a conducting wire carrying an electric current with two insulating wires which carry opposite charges and which slide over each other. Let $\lambda$ be the charge density on the positive wire assumed to be stationary and $\lambda _{-}$ the charge density on the negative wire which goes at speed $v$ . The current $I=\lambda _{-}v$ .

The negative wire represents the conduction electrons, the positive wire represents the rest of the metal wire. In the frame of reference R where the metal wire is at rest, it is electrically neutral. So in this frame $\lambda _{-}=-\lambda$ .

Let R' be a frame of reference which goes at speed $v$ with respect to R, $\lambda '_{-}$ and $\lambda '_{+}$ the densities of negative and positive charge, measured in R'. The negative wire is therefore at rest in R'.

$\lambda '_{+}=\gamma \lambda _{+}$ , because from the point of view of R', the positive wire is contracted in the direction of its movement. $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ is the inverse of the length contraction coefficient.

$\lambda _{-}=\gamma \lambda '_{-}$ , because from the point of view of R, the negative wire is contracted in the direction of its movement.

So $\lambda '_{-}\neq -\lambda '_{+}$ .

From the point of view of R', the superposition of the two wires, positive and negative, is not electrically neutral, it is therefore the source of an electrostatic field.

Consider a charge $q$ of speed $v$ in R at distance $a$ from the wire. The force F₊ exerted by the positive wire on the charge $q$ is perpendicular to the wire:

Measured in R,

$F_{+}={\frac {\lambda _{+}q}{2\pi a\epsilon _{0}}}$

This result is proven from Gauss' theorem in the chapter on Maxwell's equations.

The charge $q$ is at rest in R'. The electrostatic force $F'_{-}$ exerted by the negative wire on the charge $q$ is perpendicular to the wire.

$F'_{-}={\frac {\lambda '_{-}q}{2\pi a\epsilon _{0}}}=-{\frac {\lambda _{-}q}{2\pi a\epsilon _{0}\gamma }}$

Measured in R the force $F_{-}$ of the negative wire on the charge $q$ is equal to $F'_{-}/\gamma$ . The charge $q$ therefore experiences a force

$F_{+}+F_{-}={\frac {\lambda _{+}q(1-1/\gamma ^{2})}{2\pi a\epsilon _{0}}}={\frac {\lambda _{+}qv^{2}}{2\pi a\epsilon _{0}c^{2}}}=-{\frac {Iqv}{2\pi a\epsilon _{0}c^{2}}}$

If we set $B=-{\frac {I}{2\pi a\epsilon _{0}c^{2}}}$ we obtain

$F_{L}=F_{+}+F_{-}=qvB$

$B$ is the magnitude of the magnetic field $\mathbf {B}$ created by the current $I$ in a wire of infinite length. The Lorentz force $F_{L}=qvB$ is the magnetic force exerted on a charge $q$ which moves at speed $v$ in a magnetic field $\mathbf {B}$ if its speed is perpendicular to the field.

We thus find that the magnetic field only acts on moving electric charges.

In the frame of reference R there is no electrostatic force, because the wire carrying a current is electrically neutral. But in the frame of reference R' the negative charge density is lower than the positive charge density, because of the Fitzgerald contraction. The charge density is not zero and is the source of an electrostatic force field. This electrostatic force in R' is the magnetic force in R.

The Fitzgerald contraction and the Coulomb force are therefore sufficient to explain the existence of the magnetic force produced by an electric current.

A wire carrying a current repels negative charges that go in the conventional direction of the current (a current of positive charges) and attracts negative charges that go in the opposite direction. We thus find Ampère's law: two parallel wires carrying a current attract each other if the currents go in the same direction and repel each other if the currents go in the opposite direction.

Reference: Feynman's physics course, Electromagnetism, chapter 13-6.

The vector product

To calculate the magnetic forces produced by moving charges and the magnetic forces exerted on moving charges, one must know the cross product of two vectors in three-dimensional space:

The vector product w = u×v of two vectors u and v is the vector

whose length is uv sin $\theta$ where $\theta$ is the angle between u and v, u and v are the lengths of u and v,

whose direction is perpendicular to u and v

such that the triplet u, v, w is positively oriented.

The triplet (thumb, index, middle finger) of the right hand is positively oriented. The one of the left hand is negatively oriented. (right, front, above) is positively oriented. In general, the x, y and z axes of a coordinate system are chosen positively oriented.

The cross product of two vectors with the same direction is the zero vector. The length of the cross product of two perpendicular vectors is the product of their lengths.

The Biot-Savart law

If the regime is stationary (the electric currents do not change) the magnetic field $\mathbf {B}$ produced at the point $\mathbf {r}$ by a current $I$ in a segment $\mathbf {d\ell }$ of an electric wire is

$\mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {I\mathbf {d\ell } \times \mathbf {\hat {r}} }{r^{2}}}$

where $\mathbf {r}$ is the vector which goes from the segment $\mathbf {d\ell }$ to the point considered, $r$ is its length, $\mathbf {\hat {r}} =\mathbf {r} /r$ is the unit length vector in the direction of $\mathbf {r}$ and $\mu _{0}$ is a constant which depends on the choice of units of measurement.

With the Biot-Savart law, we can calculate the magnetic field produced by an electric current. If the wire is straight and of infinite length, the lines of the magnetic field are circles centered on the wire:

The Biot-Savart law has a mathematical form similar to that of Coulomb's law. Force is inversely proportional to the square of the distance. It is not a coincidence. We can deduce the Biot-Savart law from Coulomb's law, because the magnetic force of electric current has an electrostatic origin.

The Lorentz force

An electric field $\mathbf {E}$ and a magnetic field $\mathbf {B}$ exert on a particle of charge $q$ a force $\mathbf {F} _{L}=q(\mathbf {E} +\mathbf {v} \times \mathbf {B}$ ).

$\mathbf {F} _{L}$ is the Lorentz force:

With the Lorentz force, we explain the magnetic force on an electric current:

The principle of an electric motor:

If a current loop is parallel to a uniform magnetic field, the Lorentz force on the moving charges produces a torque that turns the loop.

The Lorentz force equation and Maxwell's equations are the fundamental laws of electromagnetism. Maxwell's equations tell how charges produce electric and magnetic fields throughout space and how these fields change over time. The Lorentz equation then tells how these fields act on charges.

The Biot-Savart law allows us to calculate the magnetic field produced by an electric current. The Lorentz force then makes it possible to calculate the force between two wires carrying a current. The magnetic field is like a mathematical intermediary for calculating the forces between moving electric charges. But it is more than a simple mathematical intermediary, because it has an autonomous existence.

We can calculate the magnetic force produced by a current from the Biot-Savart law and from Coulomb's law. The equality of the two results shows that

$\epsilon _{0}\mu _{0}=1/c^{2}$

$\epsilon _{0}$ and $\mu _{0}$ are physical constants that have been measured independently of the speed $c$ of light. When Maxwell discovered the fundamental laws of the electromagnetic field, he discovered that $c={\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}$ is the speed of electromagnetic waves. Since $c$ is also the speed of light, he concluded that light is an electromagnetic wave.

The electromotive force of magnetism

The magnetic force on a charged particle is always perpendicular to its velocity. The work of force is therefore always zero. The kinetic energy of the particle is not changed, only its direction. So how can magnetic forces produce an electric current? How can they set in motion charges initially at rest?

If a particle moves in the field created by a stationary magnet, it experiences a magnetic force. But in a frame of reference where the particle is at rest, it does not experience any magnetic force, since its speed is zero. It can only be subjected to an electrical force. So a moving magnet is the source of electrical forces and can thus produce an electric current.

If we place a conducting loop in a constant magnetic field and rotate it around a diameter perpendicular to the magnetic field, the Lorentz force sets the electrons in motion in the direction of the conducting wire, as soon as the rotation of the loop imposes on them a movement which is not parallel to the magnetic field. This movement of electrons in the direction of the wire manifests the presence of an electrical voltage. We can therefore make an electric generator by rotating a conductive loop in a constant magnetic field:

The blue area is proportional to the flux of the magnetic field through the loop. Faraday's law, presented in the chapter on Maxwell's equations, says that the voltage across the two terminals of the loop is the opposite of the rate of change of the flux of the magnetic field through the loop.

Like the magnetic force of electric current, the electromotive force of magnetism has an electrostatic origin.

To understand it, we just need to think about a square circuit traversed by a current. Let $\lambda$ be the linear density of the conduction electrons. The intensity of the current is $\lambda v$ where $v$ is the average speed of the electrons in the frame R where the circuit is at rest. Let $q$ be an electric charge placed in the center of the circuit with speed $v$ relative to R, in the direction of one of the sides of the square traveled by a current. Let R' be a frame of reference such that the charge $q$ y at rest. From the point of view of R', the charge $q$ cannot experience a magnetic force, because its speed is zero, but it experiences an electrostatic force.

From the point of view of R', the two sides of the square parallel to its movement are contracted in the direction of the movement, but not the two sides perpendicular to the movement. The linear density of the conduction electrons, on the side where their average speed is zero, is equal to $\lambda /\gamma$ where $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ is the inverse of the length contraction coefficient. The linear density of conduction electrons on the other side is $\lambda \gamma$ . It is different, because the conduction electrons have a non-zero average speed compared to R'.

From the point of view of R', the charge $q$ experiences an electrostatic force, perpendicular to the movement of the circuit and proportional to $\lambda (\gamma -1/\gamma )$ . From the point of view of R, this force is $\gamma$ times smaller, therefore proportional to $\lambda (1-1/\gamma ^{2})=\lambda v^{2}/c^{2}$ . It is the magnetic force experienced by the moving charge in R.

In R', the electric circuit is a variable magnetic field source that exerts an electric force on the charge $q$ . The variation of the magnetic field therefore has an electromotive force.