Magnetic moment

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https://en.wikipedia.org/wiki/Magnetic_moment 

In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc).

More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, the component of the magnetic moment that can be represented by an equivalent magnetic dipole: a magnetic north and south pole separated by a very small distance. The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Higher-order terms (such as the magnetic quadrupole moment) may be needed in addition to the dipole moment for extended objects.

The magnetic dipole moment of an object is readily defined in terms of the torque that the object experiences in a given magnetic field. The same applied magnetic field creates larger torques on objects with larger magnetic moments. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. The magnetic moment may be considered, therefore, to be a vector. The direction of the magnetic moment points from the south to north pole of the magnet (inside the magnet).

The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object.

Definition, units, and measurement

Definition

The magnetic moment can be defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by:

${\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }$

where τ is the torque acting on the dipole, B is the external magnetic field, and m is the magnetic moment.

This definition is based on how one could, in principle, measure the magnetic moment of an unknown sample. For a current loop, this definition leads to the magnitude of the magnetic dipole moment equaling the product of the current times the area of the loop. Further, this definition allows the calculation of the expected magnetic moment for any known macroscopic current distribution.

An alternative definition is useful for thermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy, U_int, with respect to external magnetic field:

${\displaystyle \mathbf {m} =-{\hat {\mathbf {x} }}{\frac {\partial U_{\rm {int}}}{\partial B_{x}}}-{\hat {\mathbf {y} }}{\frac {\partial U_{\rm {int}}}{\partial B_{y}}}-{\hat {\mathbf {z} }}{\frac {\partial U_{\rm {int}}}{\partial B_{z}}}.}$

Generically, the intrinsic energy includes the self-field energy of the system plus the energy of the internal workings of the system. For example, for a hydrogen atom in a 2p state in an external field, the self-field energy is negligible, so the internal energy is essentially the eigenenergy of the 2p state, which includes Coulomb potential energy and the kinetic energy of the electron. The interaction-field energy between the internal dipoles and external fields is not part of this internal energy.

Units

The unit for magnetic moment in International System of Units (SI) base units is A⋅m², where A is ampere (SI base unit of current) and m is meter (SI base unit of distance). This unit has equivalents in other SI derived units including:

${\displaystyle {\text{A}}{\cdot }{\text{m}}^{2}={\frac {{\text{N}}{\cdot }{\text{m}}}{\text{T}}}={\frac {\text{J}}{\text{T}}},}$

where N is newton (SI derived unit of force), T is tesla (SI derived unit of magnetic flux density), and J is joule (SI derived unit of energy). Although torque (N·m) and energy (J) are dimensionally equivalent, torques are never expressed in units of energy.

In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment:

${\displaystyle 1{\text{ statA}}{\cdot }{\text{cm}}^{2}=3.33564095\times 10^{-14}{\text{ A}}{\cdot }{\text{m}}^{2}}$ (ESU)

${\displaystyle 1\;{\frac {\text{erg}}{\text{G}}}=10^{-3}{\text{ A}}{\cdot }{\text{m}}^{2}}$ (Gaussian and EMU),

where statA is statamperes, cm is centimeters, erg is ergs, and G is gauss. The ratio of these two non-equivalent CGS units (EMU/ESU) is equal to the speed of light in free space, expressed in cm⋅s⁻¹.

All formulae in this article are correct in SI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with current I and area A has magnetic moment IA (see below), but in Gaussian units the magnetic moment is IA/c.

Other units for measuring the magnetic dipole moment include the Bohr magneton and the nuclear magneton.

Measurement

Main article: magnetometer

The magnetic moments of objects are typically measured with devices called magnetometers, though not all magnetometers measure magnetic moment: Some are configured to measure magnetic field instead. If the magnetic field surrounding an object is known well enough, though, then the magnetic moment can be calculated from that magnetic field.

Relation to magnetization

Main article: Magnetization

See also: Remanence

The magnetic moment is a quantity that describes the magnetic strength of an entire object. Sometimes, though, it is useful or necessary to know how much of the net magnetic moment of the object is produced by a particular portion of that magnet. Therefore, it is useful to define the magnetization field M as:

${\displaystyle \mathbf {M} ={\frac {\mathbf {m} _{\Delta V}}{V_{\Delta V}}},}$

where m_ΔV and V_ΔV are the magnetic dipole moment and volume of a sufficiently small portion of the magnet ΔV. This equation is often represented using derivative notation such that

${\displaystyle \mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}},}$

where dm is the elementary magnetic moment and dV is the volume element. The net magnetic moment of the magnet m therefore is

${\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V,}$

where the triple integral denotes integration over the volume of the magnet. For uniform magnetization (where both the magnitude and the direction of M is the same for the entire magnet (such as a straight bar magnet) the last equation simplifies to:

${\displaystyle \mathbf {m} =\mathbf {M} V,}$

where V is the volume of the bar magnet.

The magnetization is often not listed as a material parameter for commercially available ferromagnetic materials, though. Instead the parameter that is listed is residual flux density (or remanence), denoted B_r. The formula needed in this case to calculate m in (units of A⋅m²) is:

${\displaystyle \mathbf {m} ={\frac {1}{\mu _{0}}}\mathbf {B} _{\rm {r}}V}$,

where:

• B_r is the residual flux density, expressed in teslas.
• V is the volume of the magnet (in m³).
• μ₀ is the permeability of vacuum (4π×10⁻⁷ H/m).

Models

The preferred classical explanation of a magnetic moment has changed over time. Before the 1930s, textbooks explained the moment using hypothetical magnetic point charges. Since then, most have defined it in terms of Ampèrian currents. In magnetic materials, the cause of the magnetic moment are the spin and orbital angular momentum states of the electrons, and varies depending on whether atoms in one region are aligned with atoms in another.

Magnetic pole model

An electrostatic analog for a magnetic moment: two opposing charges separated by a finite distance.

The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. This is sometimes known as the Gilbert model. In this model, a small magnet is modeled by a pair of fictitious magnetic monopoles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (magnetic pole strength), and the vector ${\displaystyle \mathrm {\boldsymbol {\ell }} }$ separating them. The magnetic dipole moment m is related to the fictitious poles as

$${\displaystyle \mathbf {m} =p\,\mathrm {\boldsymbol {\ell }} \,.}$$

It points in the direction from South to North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated with angular momentum (see Relation to angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets. Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field H, in analogy to the electric field E.

Amperian loop model

The Amperian loop model: A current loop (ring) that goes into the page at the x and comes out at the dot produces a B-field (lines). The north pole is to the right and the south to the left.

After Hans Christian Ørsted discovered that electric currents produce a magnetic field and André-Marie Ampère discovered that electric currents attract and repel each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampère, the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of current I. The dipole moment of this loop is

$${\displaystyle \mathbf {m} =I{\boldsymbol {S}},}$$

where S is the area of the loop. The direction of the magnetic moment is in a direction normal to the area enclosed by the current consistent with the direction of the current using the right hand rule.

Localized current distributions

Moment ${\displaystyle {\boldsymbol {\mu }}}$ of a planar current having magnitude I and enclosing an area S

The magnetic dipole moment can be calculated for a localized (does not extend to infinity) current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from a multipole expansion of the vector potential. This leads to the definition of the magnetic dipole moment as:

$${\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\mathbf {r} \times \mathbf {j} \,{\rm {d}}V,}$$

where × is the vector cross product, r is the position vector, and j is the electric current density and the integral is a volume integral. When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then the volume integral becomes a line integral and the resulting dipole moment becomes

$${\displaystyle \mathbf {m} =I\mathbf {S} ,}$$

which is how the magnetic dipole moment for an Amperian loop is derived.

Practitioners using the current loop model generally represent the magnetic field by the solenoidal field B, analogous to the electrostatic field D.

Magnetic moment of a solenoid

Image of a solenoid

A generalization of the above current loop is a coil, or solenoid. Its moment is the vector sum of the moments of individual turns. If the solenoid has N identical turns (single-layer winding) and vector area S,

$${\displaystyle \mathbf {m} =NI\mathbf {S} .}$$

Quantum mechanical model

When calculating the magnetic moments of materials or molecules on the microscopic level it is often convenient to use a third model for the magnetic moment that exploits the linear relationship between the angular momentum and the magnetic moment of a particle. While this relation is straight forward to develop for macroscopic currents using the amperian loop model (see below), neither the magnetic pole model nor the amperian loop model truly represents what is occurring at the atomic and molecular levels. At that level quantum mechanics must be used. Fortunately, the linear relationship between the magnetic dipole moment of a particle and its angular momentum still holds, although it is different for each particle. Further, care must be used to distinguish between the intrinsic angular momentum (or spin) of the particle and the particle's orbital angular momentum. See below for more details.

Effects of an external magnetic field

Torque on a moment

The torque τ on an object having a magnetic dipole moment m in a uniform magnetic field B is:

${\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }$.

This is valid for the moment due to any localized current distribution provided that the magnetic field is uniform. For non-uniform B the equation is also valid for the torque about the center of the magnetic dipole provided that the magnetic dipole is small enough.

An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as the Larmor frequency. See Resonance.

Force on a moment

See also: Force between magnets

A magnetic moment in an externally produced magnetic field has a potential energy U:

${\displaystyle U=-\mathbf {m} \cdot \mathbf {B} }$

In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself. There are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or two monopoles (analogous to the electric dipole). The force obtained in the case of a current loop model is

${\displaystyle \mathbf {F} _{\text{loop}}=\nabla \left(\mathbf {m} \cdot \mathbf {B} \right)}$.

Assuming existence of magnetic monopole, the force is modified as follows:

${\displaystyle {\begin{aligned}\mathbf {F} _{\text{loop}}=&\left(\mathbf {m} \times \nabla \right)\times \mathbf {B} \\=&\nabla \left(\mathbf {m} \cdot \mathbf {B} \right)-\left(\nabla \cdot \mathbf {B} \right)\mathbf {m} \end{aligned}}}$

In the case of a pair of monopoles being used (i.e. electric dipole model), the force is

${\displaystyle \mathbf {F} _{\text{dipole}}=\left(\mathbf {m} \cdot \nabla \right)\mathbf {B} }$.

And one can be put in terms of the other via the relation

${\displaystyle \mathbf {F} _{\text{loop}}=\mathbf {F} _{\text{dipole}}+\mathbf {m} \times \left(\nabla \times \mathbf {B} \right)-\left(\nabla \cdot \mathbf {B} \right)\mathbf {m} }$.

In all these expressions m is the dipole and B is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields or magnetic charge, ∇×B = 0, ∇·B = 0 and the two expressions agree.

Relation to Free Energy

One can relate the magnetic moment of a system to the free energy of that system. In a uniform magnetic field B, the free energy F can be related to the magnetic moment M of the system as

${\displaystyle dF=-S\,dT-\mathbf {M} \,\cdot d\mathbf {B} }$

where S is the entropy of the system and T is the temperature. Therefore, the magnetic moment can also be defined in terms of the free energy of a system as

${\displaystyle m=\left.-\left({\frac {\partial F}{\partial B}}\right)\right|_{T}}$.

Magnetism

In addition, an applied magnetic field can change the magnetic moment of the object itself; for example by magnetizing it. This phenomenon is known as magnetism. An applied magnetic field can flip the magnetic dipoles that make up the material causing both paramagnetism and ferromagnetism. Additionally, the magnetic field can affect the currents that create the magnetic fields (such as the atomic orbits) which causes diamagnetism.

Effects on environment

Magnetic field of a magnetic moment

Main article: Magnetic dipole

See also: Dipole

 

Magnetic field lines around a "magnetostatic dipole". The magnetic dipole itself is located in the center of the figure, seen from the side, and pointing upward.

Any system possessing a net magnetic dipole moment m will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higher-order multipole components, those will drop off with distance more rapidly, so that only the dipole component will dominate the magnetic field of the system at distances far away from it.

The magnetic field of a magnetic dipole depends on the strength and direction of a magnet's magnetic moment ${\displaystyle \mathbf {m} }$ but drops off as the cube of the distance such that:

${\displaystyle {\mathbf {H} }({\mathbf {r} })={\frac {1}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{|\mathbf {r} |^{5}}}-{\frac {\mathbf {m} }{|\mathbf {r} |^{3}}}\right),}$

where ${\displaystyle \mathbf {H} }$ is the magnetic field produced by the magnet and ${\displaystyle \mathbf {r} }$ is a vector from the center of the magnetic dipole to the location where the magnetic field is measured. The inverse cube nature of this equation is more readily seen by expressing the location vector ${\displaystyle \mathbf {r} }$ as the product of its magnitude times the unit vector in its direction (${\displaystyle \mathbf {r} =|\mathbf {r} |\mathbf {\hat {r}} }$) so that:

${\displaystyle \mathbf {H} (\mathbf {r} )={\frac {1}{4\pi }}{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}.}$

The equivalent equations for the magnetic ${\displaystyle \mathbf {B} }$-field are the same except for a multiplicative factor of μ₀ = 4π×10⁻⁷ H/m, where μ₀ is known as the vacuum permeability. For example:

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}.}$

Forces between two magnetic dipoles

See also: Magnetic dipole–dipole interaction

As discussed earlier, the force exerted by a dipole loop with moment m₁ on another with moment m₂ is

${\displaystyle \mathbf {F} =\nabla \left(\mathbf {m} _{2}\cdot \mathbf {B} _{1}\right),}$

where B₁ is the magnetic field due to moment m₁. The result of calculating the gradient is

${\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\frac {3\mu _{0}}{4\pi |\mathbf {r} |^{4}}}\left(\mathbf {m} _{2}(\mathbf {m} _{1}\cdot {\hat {\mathbf {r} }})+\mathbf {m} _{1}(\mathbf {m} _{2}\cdot {\hat {\mathbf {r} }})+{\hat {\mathbf {r} }}(\mathbf {m} _{1}\cdot \mathbf {m} _{2})-5{\hat {\mathbf {r} }}(\mathbf {m} _{1}\cdot {\hat {\mathbf {r} }})(\mathbf {m} _{2}\cdot {\hat {\mathbf {r} }})\right),}$

where r̂ is the unit vector pointing from magnet 1 to magnet 2 and r is the distance. An equivalent expression is

${\displaystyle \mathbf {F} ={\frac {3\mu _{0}}{4\pi |\mathbf {r} |^{4}}}\left(({\hat {\mathbf {r} }}\times \mathbf {m} _{1})\times \mathbf {m} _{2}+({\hat {\mathbf {r} }}\times \mathbf {m} _{2})\times \mathbf {m} _{1}-2{\hat {\mathbf {r} }}(\mathbf {m} _{1}\cdot \mathbf {m} _{2})+5{\hat {\mathbf {r} }}({\hat {\mathbf {r} }}\times \mathbf {m} _{1})\cdot ({\hat {\mathbf {r} }}\times \mathbf {m} _{2})\right).}$

The force acting on m₁ is in the opposite direction.

Torque of one magnetic dipole on another

The torque of magnet 1 on magnet 2 is

${\displaystyle {\boldsymbol {\tau }}=\mathbf {m} _{2}\times \mathbf {B} _{1}.}$

Theory underlying magnetic dipoles

Main article: Multipole expansion

See also: Magnetic dipole

The magnetic field of any magnet can be modeled by a series of terms for which each term is more complicated (having finer angular detail) than the one before it. The first three terms of that series are called the monopole (represented by an isolated magnetic north or south pole) the dipole (represented by two equal and opposite magnetic poles), and the quadrupole (represented by four poles that together form two equal and opposite dipoles). The magnitude of the magnetic field for each term decreases progressively faster with distance than the previous term, so that at large enough distances the first non-zero term will dominate.

For many magnets the first non-zero term is the magnetic dipole moment. (To date, no isolated magnetic monopoles have been experimentally detected.) A magnetic dipole is the limit of either a current loop or a pair of poles as the dimensions of the source are reduced to zero while keeping the moment constant. As long as these limits only apply to fields far from the sources, they are equivalent. However, the two models give different predictions for the internal field (see below).

Magnetic potentials

Traditionally, the equations for the magnetic dipole moment (and higher order terms) are derived from theoretical quantities called magnetic potentials which are simpler to deal with mathematically than the magnetic fields.

In the magnetic pole model, the relevant magnetic field is the demagnetizing field ${\displaystyle \mathbf {H} }$. Since the demagnetizing portion of ${\displaystyle \mathbf {H} }$ does not include, by definition, the part of ${\displaystyle \mathbf {H} }$ due to free currents, there exists a magnetic scalar potential such that

${\displaystyle {\mathbf {H} }({\mathbf {r} })=-\nabla \psi }$.

In the amperian loop model, the relevant magnetic field is the magnetic induction ${\displaystyle \mathbf {B} }$. Since magnetic monopoles do not exist, there exists a magnetic vector potential such that

${\displaystyle \mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }.}$

Both of these potentials can be calculated for any arbitrary current distribution (for the amperian loop model) or magnetic charge distribution (for the magnetic charge model) provided that these are limited to a small enough region to give:

${\displaystyle {\begin{aligned}\mathbf {A} \left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {j} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} V',\\\psi \left(\mathbf {r} ,t\right)&={\frac {1}{4\pi }}\int {\frac {\rho \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} V',\end{aligned}}}$

where ${\displaystyle \mathbf {j} }$ is the current density in the amperian loop model, ${\displaystyle \rho }$ is the magnetic pole strength density in analogy to the electric charge density that leads to the electric potential, and the integrals are the volume (triple) integrals over the coordinates that make up ${\displaystyle \mathbf {r} '}$. The denominators of these equation can be expanded using the multipole expansion to give a series of terms that have larger of power of distances in the denominator. The first nonzero term, therefore, will dominate for large distances. The first non-zero term for the vector potential is:

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{|\mathbf {r} |^{3}}},}$

where ${\displaystyle \mathbf {m} }$ is:

${\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\mathbf {r} \times \mathbf {j} \,{\rm {d}}V,}$

where × is the vector cross product, r is the position vector, and j is the electric current density and the integral is a volume integral.

In the magnetic pole perspective, the first non-zero term of the scalar potential is

${\displaystyle \psi (\mathbf {r} )={\frac {\mathbf {m} \cdot \mathbf {r} }{4\pi |\mathbf {r} |^{3}}}.}$

Here ${\displaystyle \mathbf {m} }$ may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of the magnetization field as:

${\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V.}$

The same symbol ${\displaystyle \mathbf {m} }$ is used for both equations since they produce equivalent results outside of the magnet.

External magnetic field produced by a magnetic dipole moment

The magnetic flux density for a magnetic dipole in the amperian loop model, therefore, is

${\displaystyle \mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{|\mathbf {r} |^{5}}}-{\frac {\mathbf {m} }{|\mathbf {r} |^{3}}}\right).}$

Further, the magnetic field strength ${\displaystyle \mathbf {H} }$ is

${\displaystyle {\mathbf {H} }({\mathbf {r} })=-\nabla \psi ={\frac {1}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{|\mathbf {r} |^{5}}}-{\frac {\mathbf {m} }{|\mathbf {r} |^{3}}}\right).}$

Internal magnetic field of a dipole

The magnetic field of a current loop

The two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field far from the source. However, inside the source region, they give different predictions. The magnetic field between poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). The limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.

If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}+{\frac {8\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right].}$

Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.

If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole charge and distance constant, the limiting field is

${\displaystyle \mathbf {H} (\mathbf {r} )={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}-{\frac {4\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right].}$

These fields are related by B = μ₀(H + M), where M(r) = mδ(r) is the magnetization.

Relation to angular momentum

See also: gyromagnetic ratio

The magnetic moment has a close connection with angular momentum called the gyromagnetic effect. This effect is expressed on a macroscopic scale in the Einstein–de Haas effect, or "rotation by magnetization," and its inverse, the Barnett effect, or "magnetization by rotation." Further, a torque applied to a relatively isolated magnetic dipole such as an atomic nucleus can cause it to precess (rotate about the axis of the applied field). This phenomenon is used in nuclear magnetic resonance.

Viewing a magnetic dipole as current loop brings out the close connection between magnetic moment and angular momentum. Since the particles creating the current (by rotating around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called the gyromagnetic ratio or ${\displaystyle \gamma }$ so that:

${\displaystyle \mathbf {m} =\gamma \,\mathbf {L} ,}$

where ${\displaystyle \mathbf {L} }$ is the angular momentum of the particle or particles that are creating the magnetic moment.

In the amperian loop model, which applies for macroscopic currents, the gyromagnetic ratio is one half of the charge-to-mass ratio. This can be shown as follows. The angular momentum of a moving charged particle is defined as:

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mu \,\mathbf {r} \times \mathbf {v} ,}$

where μ is the mass of the particle and v is the particle's velocity. The angular momentum of the very large number of charged particles that make up a current therefore is:

${\displaystyle \mathbf {L} =\iiint _{V}\,\mathbf {r} \times (\rho \mathbf {v} )\,{\rm {d}}V\,,}$

where ρ is the mass density of the moving particles. By convention the direction of the cross product is given by the right-hand rule.

This is similar to the magnetic moment created by the very large number of charged particles that make up that current:

${\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\,\mathbf {r} \times (\rho _{Q}\mathbf {v} )\,{\rm {d}}V\,,}$

where ${\displaystyle \mathbf {j} =\rho _{Q}\mathbf {v} }$ and ${\displaystyle \rho _{Q}}$ is the charge density of the moving charged particles.

Comparing the two equations results in:

${\displaystyle \mathbf {m} ={\frac {e}{2\mu }}\,\mathbf {L} \,,}$

where ${\displaystyle e}$ is the charge of the particle and ${\displaystyle \mu }$ is the mass of the particle.

Even though atomic particles cannot be accurately described as orbiting (and spinning) charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world so that:

${\displaystyle \mathbf {m} =g\,{\frac {e}{2\mu }}\,\mathbf {L} ,}$

where the g-factor depends on the particle and configuration. For example the g-factor for the magnetic moment due to an electron orbiting a nucleus is one while the g-factor for the magnetic moment of electron due to its intrinsic angular momentum (spin) is a little larger than 2. The g-factor of atoms and molecules must account for the orbital and intrinsic moments of its electrons and possibly the intrinsic moment of its nuclei as well.

In the atomic world the angular momentum (spin) of a particle is an integer (or half-integer in the case of spin) multiple of the reduced Planck constant ħ. This is the basis for defining the magnetic moment units of Bohr magneton (assuming charge-to-mass ratio of the electron) and nuclear magneton (assuming charge-to-mass ratio of the proton). See electron magnetic moment and Bohr magneton for more details.

Atoms, molecules, and elementary particles

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: motion of electric charges, such as electric currents; and the intrinsic magnetism of elementary particles, such as the electron.

Contributions due to the sources of the first kind can be calculated from knowing the distribution of all the electric currents (or, alternatively, of all the electric charges and their velocities) inside the system, by using the formulas below. On the other hand, the magnitude of each elementary particle's intrinsic magnetic moment is a fixed number, often measured experimentally to a great precision. For example, any electron's magnetic moment is measured to be −9.284764×10⁻²⁴ J/T. The direction of the magnetic moment of any elementary particle is entirely determined by the direction of its spin, with the negative value indicating that any electron's magnetic moment is antiparallel to its spin.

The net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, the magnetic moment of an atom of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions:

1. the intrinsic moment of the electron,
2. the orbital motion of the electron around the proton,
3. the intrinsic moment of the proton.

Similarly, the magnetic moment of a bar magnet is the sum of the contributing magnetic moments, which include the intrinsic and orbital magnetic moments of the unpaired electrons of the magnet's material and the nuclear magnetic moments.

Magnetic moment of an atom

For an atom, individual electron spins are added to get a total spin, and individual orbital angular momenta are added to get a total orbital angular momentum. These two then are added using angular momentum coupling to get a total angular momentum. For an atom with no nuclear magnetic moment, the magnitude of the atomic dipole moment, ${\displaystyle {\mathfrak {m}}_{\text{atom}}}$, is then

${\displaystyle {\mathfrak {m}}_{\text{atom}}=g_{\rm {J}}\,\mu _{\rm {B}}\,{\sqrt {j\,(j+1)\,}}}$

where j is the total angular momentum quantum number, g_J is the Landé g-factor, and μ_B is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then

$${\displaystyle {\mathfrak {m}}_{{\text{atom}},z}=-m\,g_{\rm {J}}\,\mu _{\rm {B}}~.}$$

The negative sign occurs because electrons have negative charge.

The integer m (not to be confused with the moment, ${\displaystyle {\mathfrak {m}}}$) is called the magnetic quantum number or the equatorial quantum number, which can take on any of 2j + 1 values:

$${\displaystyle -j,\ -(j-1),\ \cdots ,\ -1,\ 0,\ +1,\ \cdots ,\ +(j-1),\ +j~.}$$

Due to the angular momentum, the dynamics of a magnetic dipole in a magnetic field differs from that of an electric dipole in an electric field. The field does exert a torque on the magnetic dipole tending to align it with the field. However, torque is proportional to rate of change of angular momentum, so precession occurs: the direction of spin changes. This behavior is described by the Landau–Lifshitz–Gilbert equation:

${\displaystyle {\frac {1}{\gamma }}{\frac {{\rm {d}}\mathbf {m} }{{\rm {d}}t}}=\mathbf {m} \times \mathbf {H} _{\text{eff}}-{\frac {\lambda }{\gamma m}}\mathbf {m} \times {\frac {{\rm {d}}\mathbf {m} }{{\rm {d}}t}}}$

where γ is the gyromagnetic ratio, m is the magnetic moment, λ is the damping coefficient and H_eff is the effective magnetic field (the external field plus any self-induced field). The first term describes precession of the moment about the effective field, while the second is a damping term related to dissipation of energy caused by interaction with the surroundings.

Magnetic moment of an electron

See also: Anomalous magnetic dipole moment

Electrons and many elementary particles also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentum of the particles as discussed in the article Electron magnetic moment. It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as electron paramagnetic resonance.

The magnetic moment of the electron is

${\displaystyle \mathbf {m} _{\text{S}}=-{\frac {g_{\text{S}}\mu _{\text{B}}\mathbf {S} }{\hbar }},}$

where μ_B is the Bohr magneton, S is electron spin, and the g-factor g_S is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: 2.00231930436. The deviation from 2 is known as the anomalous magnetic dipole moment.

Again it is important to notice that m is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. This can be understood with the following classical picture: if we imagine that the spin angular momentum is created by the electron mass spinning around some axis, the electric current that this rotation creates circulates in the opposite direction, because of the negative charge of the electron; such current loops produce a magnetic moment which is antiparallel to the spin. Hence, for a positron (the anti-particle of the electron) the magnetic moment is parallel to its spin.

Magnetic moment of a nucleus

See also: Nuclear magnetic moment

The nuclear system is a complex physical system consisting of nucleons, i.e., protons and neutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depend on the spin of the individual nucleons, one can look at these properties with measurements of nuclear moments, and more specifically the nuclear magnetic dipole moment.

Most common nuclei exist in their ground state, although nuclei of some isotopes have long-lived excited states. Each energy state of a nucleus of a given isotope is characterized by a well-defined magnetic dipole moment, the magnitude of which is a fixed number, often measured experimentally to a great precision. This number is very sensitive to the individual contributions from nucleons, and a measurement or prediction of its value can reveal important information about the content of the nuclear wave function. There are several theoretical models that predict the value of the magnetic dipole moment and a number of experimental techniques aiming to carry out measurements in nuclei along the nuclear chart.

Magnetic moment of a molecule

Any molecule has a well-defined magnitude of magnetic moment, which may depend on the molecule's energy state. Typically, the overall magnetic moment of a molecule is a combination of the following contributions, in the order of their typical strength:

• magnetic moments due to its unpaired electron spins (paramagnetic contribution), if any
• orbital motion of its electrons, which in the ground state is often proportional to the external magnetic field (diamagnetic contribution)
• the combined magnetic moment of its nuclear spins, which depends on the nuclear spin configuration.

Examples of molecular magnetism

• The dioxygen molecule, O₂, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons.
• The carbon dioxide molecule, CO₂, mostly exhibits diamagnetism, a much weaker magnetic moment of the electron orbitals that is proportional to the external magnetic field. The nuclear magnetism of a magnetic isotope such as ¹³C or ¹⁷O will contribute to the molecule's magnetic moment.
• The dihydrogen molecule, H₂, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in a para- or an ortho- nuclear spin configuration.
• Many transition metal complexes are magnetic. The spin-only formula is a good first approximation for high-spin complexes of first-row transition metals.

Number of unpaired electrons	Spin-only moment (μB)
1	1.73
2	2.83
3	3.87
4	4.90
5	5.92

Elementary particles

In atomic and nuclear physics, the Greek symbol μ represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Values of the intrinsic magnetic moments of some particles are given in the table below:

Intrinsic magnetic moments and spins
of some elementary particles

Particle name (symbol)	Magnetic dipole moment (10−27 J⋅T−1)	Spin quantum number (dimensionless)
electron (e−)	−9284.764	1/2
proton (H+)	14.106067	1/2
neutron (n)	−9.66236	1/2
muon (μ−)	−44.904478	1/2
deuteron (2H+)	4.3307346	1
triton (3H+)	15.046094	1/2
helion (3He++)	−10.746174	1/2
alpha particle (4He++)	0	0

For the relation between the notions of magnetic moment and magnetization see magnetization.

Electromagnetic mass

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electromagnetic_mass 

Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass per se. Today, the relation of mass, momentum, velocity, and all forms of energy – including electromagnetic energy – is analyzed on the basis of Albert Einstein's special relativity and mass–energy equivalence. As to the cause of mass of elementary particles, the Higgs mechanism in the framework of the relativistic Standard Model is currently used. However, some problems concerning the electromagnetic mass and self-energy of charged particles are still studied.

Charged particles

Rest mass and energy

It was recognized by J. J. Thomson in 1881 that a charged sphere moving in a space filled with a medium of a specific inductive capacity (the electromagnetic aether of James Clerk Maxwell), is harder to set in motion than an uncharged body. (Similar considerations were already made by George Gabriel Stokes (1843) with respect to hydrodynamics, who showed that the inertia of a body moving in an incompressible perfect fluid is increased.) So due to this self-induction effect, electrostatic energy behaves as having some sort of momentum and "apparent" electromagnetic mass, which can increase the ordinary mechanical mass of the bodies, or in more modern terms, the increase should arise from their electromagnetic self-energy. This idea was worked out in more detail by Oliver Heaviside (1889), Thomson (1893), George Frederick Charles Searle (1897), Max Abraham (1902), Hendrik Lorentz (1892, 1904), and was directly applied to the electron by using the Abraham–Lorentz force. Now, the electrostatic energy ${\displaystyle E_{\mathrm {em} }}$ and mass ${\displaystyle m_{\mathrm {em} }}$ of an electron at rest was calculated to be

${\displaystyle E_{\mathrm {em} }={\frac {1}{2}}{\frac {e^{2}}{a}},\qquad m_{\mathrm {em} }={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}}$

where ${\displaystyle e}$ is the charge, uniformly distributed on the surface of a sphere, and ${\displaystyle a}$ is the classical electron radius, which must be nonzero to avoid infinite energy accumulation. Thus the formula for this electromagnetic energy–mass relation is

${\displaystyle m_{\mathrm {em} }={\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}}$

This was discussed in connection with the proposal of the electrical origin of matter, so Wilhelm Wien (1900), and Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. Wien stated, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a proportionality between electromagnetic energy, inertial mass, and gravitational mass. When one body attracts another one, the electromagnetic energy store of gravitation is according to Wien diminished by the amount (where ${\displaystyle M}$ is the attracted mass, ${\displaystyle G}$ the gravitational constant, ${\displaystyle r}$ the distance):

${\displaystyle G={\frac {{\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}M}{r}}}$

Henri Poincaré in 1906 argued that when mass is in fact the product of the electromagnetic field in the aether – implying that no "real" mass exists – and because matter is inseparably connected with mass, then also matter doesn't exist at all and electrons are only concavities in the aether.

Mass and speed

Thomson and Searle

Thomson (1893) noticed that electromagnetic momentum and energy of charged bodies, and therefore their masses, depend on the speed of the bodies as well. He wrote:

[p. 21] When in the limit v = c, the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain constant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light.

In 1897, Searle gave a more precise formula for the electromagnetic energy of charged sphere in motion:

${\displaystyle E_{\mathrm {em} }^{v}=E_{\mathrm {em} }\left[{\frac {1}{\beta }}\ln {\frac {1+\beta }{1-\beta }}-1\right],\qquad \beta ={\frac {v}{c}},}$

and like Thomson he concluded:

... when v = c the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.

Longitudinal and transverse mass

Predictions of speed dependence of transverse electromagnetic mass according to the theories of Abraham, Lorentz, and Bucherer.

From Searle's formula, Walter Kaufmann (1901) and Abraham (1902) derived the formula for the electromagnetic mass of moving bodies:

${\displaystyle m_{L}={\frac {3}{4}}\cdot m_{\mathrm {em} }\cdot {\frac {1}{\beta ^{2}}}\left[-{\frac {1}{\beta }}\ln \left({\frac {1+\beta }{1-\beta }}\right)+{\frac {2}{1-\beta ^{2}}}\right]}$

However, it was shown by Abraham (1902), that this value is only valid in the longitudinal direction ("longitudinal mass"), i.e., that the electromagnetic mass also depends on the direction of the moving bodies with respect to the aether. Thus Abraham also derived the "transverse mass":

${\displaystyle m_{T}={\frac {3}{4}}\cdot m_{\mathrm {em} }\cdot {\frac {1}{\beta ^{2}}}\left[\left({\frac {1+\beta ^{2}}{2\beta }}\right)\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right]}$

On the other hand, already in 1899 Lorentz assumed that the electrons undergo length contraction in the line of motion, which leads to results for the acceleration of moving electrons that differ from those given by Abraham. Lorentz obtained factors of ${\displaystyle k^{3}\varepsilon }$ parallel to the direction of motion and ${\displaystyle k\varepsilon }$ perpendicular to the direction of motion, where ${\displaystyle k={\sqrt {1-v^{2}/c^{2}}}}$ and ${\displaystyle \varepsilon }$ is an undetermined factor. Lorentz expanded his 1899 ideas in his famous 1904 paper, where he set the factor ${\displaystyle \varepsilon }$ to unity, thus:

${\displaystyle m_{L}={\frac {m_{\mathrm {em} }}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{\mathrm {em} }}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$,

So, eventually Lorentz arrived at the same conclusion as Thomson in 1893: no body can reach the speed of light because the mass becomes infinitely large at this velocity.

Additionally, a third electron model was developed by Alfred Bucherer and Paul Langevin, in which the electron contracts in the line of motion, and expands perpendicular to it, so that the volume remains constant. This gives:

${\displaystyle m_{L}={\frac {m_{\mathrm {em} }\left(1-{\frac {1}{3}}{\frac {v^{2}}{c^{2}}}\right)}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{8/3}}},\quad m_{T}={\frac {m_{\mathrm {em} }}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{2/3}}}}$

Kaufmann's experiments

The predictions of the theories of Abraham and Lorentz were supported by the experiments of Walter Kaufmann (1901), but the experiments were not precise enough to distinguish between them. In 1905 Kaufmann conducted another series of experiments (Kaufmann–Bucherer–Neumann experiments) which confirmed Abraham's and Bucherer's predictions, but contradicted Lorentz's theory and the "fundamental assumption of Lorentz and Einstein", i.e., the relativity principle. In the following years experiments by Alfred Bucherer (1908), Gunther Neumann (1914) and others seemed to confirm Lorentz's mass formula. It was later pointed out that the Bucherer–Neumann experiments were also not precise enough to distinguish between the theories – it lasted until 1940 when the precision required was achieved to eventually prove Lorentz's formula and to refute Abraham's by these kinds of experiments. (However, other experiments of different kind already refuted Abraham's and Bucherer's formulas long before.)

Poincaré stresses and the 4⁄3 problem

The idea of an electromagnetic nature of matter, however, had to be given up. Abraham (1904, 1905) argued that non-electromagnetic forces were necessary to prevent Lorentz's contractile electrons from exploding. He also showed that different results for the longitudinal electromagnetic mass can be obtained in Lorentz's theory, depending on whether the mass is calculated from its energy or its momentum, so a non-electromagnetic potential (corresponding to 1⁄3 of the electron's electromagnetic energy) was necessary to render these masses equal. Abraham doubted whether it was possible to develop a model satisfying all of these properties.

To solve those problems, Henri Poincaré in 1905 and 1906 introduced some sort of pressure ("Poincaré stresses") of non-electromagnetic nature. As required by Abraham, these stresses contribute non-electromagnetic energy to the electrons, amounting to 1⁄4 of their total energy or to 1⁄3 of their electromagnetic energy. So, the Poincaré stresses remove the contradiction in the derivation of the longitudinal electromagnetic mass, they prevent the electron from exploding, they remain unaltered by a Lorentz transformation (i.e. they are Lorentz invariant), and were also thought as a dynamical explanation of length contraction. However, Poincaré still assumed that only the electromagnetic energy contributes to the mass of the bodies.

As it was later noted, the problem lies in the 4⁄3 factor of electromagnetic rest mass – given above as ${\displaystyle m_{\mathrm {em} }={\tfrac {4}{3}}E_{\mathrm {em} }/c^{2}}$ when derived from the Abraham–Lorentz equations. However, when it is derived from the electron's electrostatic energy alone, we have ${\displaystyle m_{\mathrm {es} }=E_{\mathrm {em} }/c^{2}}$ where the 4⁄3 factor is missing. This can be solved by adding the non-electromagnetic energy ${\displaystyle E_{\mathrm {p} }}$ of the Poincaré stresses to ${\displaystyle E_{\mathrm {em} }}$, the electron's total energy ${\displaystyle E_{\mathrm {tot} }}$ now becomes:

${\displaystyle {\frac {E_{\mathrm {tot} }}{c^{2}}}={\frac {E_{\mathrm {em} }+E_{\mathrm {p} }}{c^{2}}}={\frac {E_{\mathrm {em} }+{\frac {E_{\mathrm {em} }}{3}}}{c^{2}}}={\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}={\frac {4}{3}}m_{\mathrm {es} }=m_{\mathrm {em} }}$

Thus the missing 4⁄3 factor is restored when the mass is related to its electromagnetic energy, and it disappears when the total energy is considered.

Inertia of energy and radiation paradoxes

Radiation pressure

Another way of deriving some sort of electromagnetic mass was based on the concept of radiation pressure. These pressures or tensions in the electromagnetic field were derived by James Clerk Maxwell (1874) and Adolfo Bartoli (1876). Lorentz recognized in 1895 that those tensions also arise in his theory of the stationary aether. So if the electromagnetic field of the aether is able to set bodies in motion, the action / reaction principle demands that the aether must be set in motion by matter as well. However, Lorentz pointed out that any tension in the aether requires the mobility of the aether parts, which is not possible since in his theory the aether is immobile. (unlike contemporaries like Thomson who used fluid descriptions) This represents a violation of the reaction principle that was accepted by Lorentz consciously. He continued by saying, that one can only speak about fictitious tensions, since they are only mathematical models in his theory to ease the description of the electrodynamic interactions.

Mass of the fictitious electromagnetic fluid

In 1900 Poincaré studied the conflict between the action/reaction principle and Lorentz's theory. He tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields and radiation are involved. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum (such a momentum was also derived by Thomson in 1893 in a more complicated way). Poincaré concluded, the electromagnetic field energy behaves like a fictitious fluid („fluide fictif“) with a mass density of ${\displaystyle E_{em}/c^{2}}$ (in other words ${\displaystyle m_{em}=E_{em}/c^{2}}$). Now, if the center of mass frame (COM-frame) is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible – it is neither created or destroyed – then the motion of the center of mass frame remains uniform.

But this electromagnetic fluid is not indestructible, because it can be absorbed by matter (which according to Poincaré was the reason why he regarded the em-fluid as "fictitious" rather than "real"). Thus the COM-principle would be violated again. As it was later done by Einstein, an easy solution of this would be to assume that the mass of em-field is transferred to matter in the absorption process. But Poincaré created another solution: He assumed that there exists an immobile non-electromagnetic energy fluid at each point in space, also carrying a mass proportional to its energy. When the fictitious em-fluid is destroyed or absorbed, its electromagnetic energy and mass is not carried away by moving matter, but is transferred into the non-electromagnetic fluid and remains at exactly the same place in that fluid. (Poincaré added that one should not be too surprised by these assumptions, since they are only mathematical fictions.) In this way, the motion of the COM-frame, including matter, fictitious em-fluid, and fictitious non-em-fluid, at least theoretically remains uniform.

However, since only matter and electromagnetic energy are directly observable by experiment (not the non-em-fluid), Poincaré's resolution still violates the reaction principle and the COM-theorem, when an emission/absorption process is practically considered. This leads to a paradox when changing frames: if waves are radiated in a certain direction, the device will suffer a recoil from the momentum of the fictitious fluid. Then, Poincaré performed a Lorentz boost (to first order in v/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether.

Poincaré came back to this topic in 1904. This time he rejected his own solution that motions in the ether can compensate the motion of matter, because any such motion is unobservable and therefore scientifically worthless. He also abandoned the concept that energy carries mass and wrote in connection to the above-mentioned recoil:

The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy.

These iterative developments culminated in his 1906 publication "The End of Matter" in which he notes that when applying the methodology of using an electric or magnetic field deviations to determine charge-to-mass ratios, one finds that the apparent mass added by charge makes up all of the apparent mass, thus the "real mass is equal to zero." Thus he goes on to postulate that electrons are only holes or motion effects in the aether while the aether itself is the only thing "endowed with inertia."

He then goes on to address the possibility that all matter might share this same quality and thereby his position changes from viewing aether as a "fictitious fluid" to suggesting it might be the only thing that actually exists in the universe, finally stating "In this system there is no actual matter, there are only holes in the aether."

Momentum and cavity radiation

However, Poincaré's idea of momentum and mass associated with radiation proved to be fruitful, when in 1903 Max Abraham introduced the term „electromagnetic momentum“, having a field density of ${\displaystyle E_{em}/c^{2}}$ per cm³ and ${\displaystyle E_{em}/c}$ per cm². Contrary to Lorentz and Poincaré, who considered momentum as a fictitious force, he argued that it is a real physical entity, and therefore conservation of momentum is guaranteed.

In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation by studying the dynamics of a moving cavity. Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that ${\displaystyle m={\tfrac {8}{3}}E/c^{2}}$. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and on the basis of his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to ${\displaystyle m={\tfrac {4}{3}}E/c^{2}}$, the same value for the electromagnetic mass for a body at rest. Hasenöhrl recalculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass that Poincaré would comment on in 1906. However, Hasenöhrl stated that this energy-apparent-mass relation only holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K.

Modern view

Mass–energy equivalence

The idea that the principal relations between mass, energy, momentum and velocity can only be considered on the basis of dynamical interactions of matter was superseded, when Albert Einstein found out in 1905 that considerations based on the special principle of relativity require that all forms of energy (not only electromagnetic) contribute to the mass of bodies (mass–energy equivalence). That is, the entire mass of a body is a measure of its energy content by ${\displaystyle E=mc^{2}}$, and Einstein's considerations were independent from assumptions about the constitution of matter. By this equivalence, Poincaré's radiation paradox can be solved without using "compensating forces", because the mass of matter itself (not the non-electromagnetic aether fluid as suggested by Poincaré) is increased or diminished by the mass of electromagnetic energy in the course of the emission/absorption process. Also the idea of an electromagnetic explanation of gravitation was superseded in the course of developing general relativity.

So every theory dealing with the mass of a body must be formulated in a relativistic way from the outset. This is for example the case in the current quantum field explanation of mass of elementary particles in the framework of the Standard Model, the Higgs mechanism. Because of this, the idea that any form of mass is completely caused by interactions with electromagnetic fields, is not relevant any more.

Relativistic mass

The concepts of longitudinal and transverse mass (equivalent to those of Lorentz) were also used by Einstein in his first papers on relativity. However, in special relativity they apply to the entire mass of matter, not only to the electromagnetic part. Later it was shown by physicists like Richard Chace Tolman that expressing mass as the ratio of force and acceleration is not advantageous. Therefore, a similar concept without direction dependent terms, in which force is defined as ${\displaystyle {\vec {F}}=\mathrm {d} {\vec {p}}/\mathrm {d} t}$, was used as relativistic mass

${\displaystyle M={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\qquad m_{0}={\frac {E}{c^{2}}},}$

This concept is sometimes still used in modern physics textbooks, although the term 'mass' is now considered by many to refer to invariant mass, see mass in special relativity.

Self-energy

When the special case of the electromagnetic self-energy or self-force of charged particles is discussed, also in modern texts some sort of "effective" electromagnetic mass is sometimes introduced – not as an explanation of mass per se, but in addition to the ordinary mass of bodies. Many different reformulations of the Abraham–Lorentz force have been derived – for instance, in order to deal with the 4⁄3-problem (see next section) and other problems that arose from this concept. Such questions are discussed in connection with renormalization, and on the basis of quantum mechanics and quantum field theory, which must be applied when the electron is considered physically point-like. At distances located in the classical domain, the classical concepts again come into play. A rigorous derivation of the electromagnetic self-force, including the contribution to the mass of the body, was published by Gralla et al. (2009).

4⁄3 problem

Max von Laue in 1911 also used the Abraham–Lorentz equations of motion in his development of special relativistic dynamics, so that also in special relativity the 4⁄3 factor is present when the electromagnetic mass of a charged sphere is calculated. This contradicts the mass–energy equivalence formula, which requires the relation ${\displaystyle m_{\mathrm {em} }=E_{\mathrm {em} }/c^{2}}$ without the 4⁄3 factor, or in other words, four-momentum doesn't properly transform like a four-vector when the 4⁄3 factor is present. Laue found a solution equivalent to Poincaré's introduction of a non-electromagnetic potential (Poincaré stresses), but Laue showed its deeper, relativistic meaning by employing and advancing Hermann Minkowski's spacetime formalism. Laue's formalism required that there are additional components and forces, which guarantee that spatially extended systems (where both electromagnetic and non-electromagnetic energies are combined) are forming a stable or "closed system" and transform as a four-vector. That is, the 4⁄3 factor arises only with respect to electromagnetic mass, while the closed system has total rest mass and energy of ${\displaystyle m_{\mathrm {tot} }=E_{\mathrm {tot} }/c^{2}}$.

Another solution was found by authors such as Enrico Fermi (1922), Paul Dirac (1938) Fritz Rohrlich (1960), or Julian Schwinger (1983), who pointed out that the electron's stability and the 4/3-problem are two different things. They showed that the preceding definitions of four-momentum are non-relativistic per se, and by changing the definition into a relativistic form, the electromagnetic mass can simply be written as ${\displaystyle m_{\mathrm {em} }=E_{\mathrm {em} }/c^{2}}$ and thus the 4⁄3 factor doesn't appear at all. So every part of the system, not only "closed" systems, properly transforms as a four-vector. However, binding forces like the Poincaré stresses are still necessary to prevent the electron from exploding due to Coulomb repulsion. But on the basis of the Fermi–Rohrlich definition, this is only a dynamical problem and has nothing to do with the transformation properties any more.

Also other solutions have been proposed, for instance, Valery Morozov (2011) gave consideration to movement of an imponderable charged sphere. It turned out that a flux of nonelectromagnetic energy exists in the sphere body. This flux has an impulse exactly equal to 1⁄3 of the sphere electromagnetic impulse regardless of a sphere internal structure or a material, it is made of. The problem was solved without attraction of any additional hypotheses. In this model, sphere tensions are not connected with its mass.