The electromagnetic mass of a system is the contribution of electromagnetic interactions to its inertia as calculated using classical electrodynamics. Different approaches to computing the electromagnetic mass give different answers. Electromagnetic mass, like inductance or the Abraham–Lorentz force, is a self-interaction phenomenon, in the sense that a charged body interacts with its own electromagnetic field. The concept first introduced in 1881 by Joseph J. Thomson.
Physical origin In classical mechanics a particle's momentum, p {\displaystyle p} is proportional to its velocity, v {\displaystyle v} :
p = m v {\displaystyle p=mv}
where the constant of proportionality is the particle's mass, m {\displaystyle m} . In classical electrodynamics a particle with charge like an electron might be modeled as a conducting sphere. As this sphere moves at non-relativistic speed in its own (radial) electric field, the interaction of the charge with the field gives an electromagnetic momentum:
p = 2 3 e 2 a c 2 v {\displaystyle p={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}v}
where e {\displaystyle e} is the elementary charge, c {\displaystyle c} is the speed of light, and a {\displaystyle a} is the radius of the particle. By comparison with the mechanical momentum, the electromagnetic mass of the spherical model of the electron can be defined:
m elec = 2 3 e 2 a c 2 {\displaystyle m_{\textrm {elec}}={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}}
However, applying the same type of analysis to the electrostatic energy of the sphere produces another value for the electromagnetic mass:
m elec ′ = 1 2 e 2 a c 2 {\displaystyle m'_{\textrm {elec}}={\frac {1}{2}}{\frac {e^{2}}{ac^{2}}}}
Nevertheless, a charged conducting sphere is unstable without some other force to counteract the repulsion, and whatever that force might be would also contribute to mass.
Derivation based on self-force A system consisting of two point electric charges q1 and q2 separated by a distance r is the simplest extended charge distribution, and it can be used to illustrate various approaches to computing the electromagnetic mass. In addition to the energy-based and momentum-based approaches, the self-force mass can be computed as follows.
When this system of two point charges is at rest, the electric force exerted by q1 on q2 is equal and opposite to that exerted by q2 on q1 (Coulomb's law). If the pair of charges is now accelerated, with an acceleration a → {\displaystyle {\vec {a}}} directed to the right (keeping the distance r between the two charges constant), this is no longer the case: since the electric field produced by each charge does not propagate instantaneously but at the speed of light (c = 299,792,458 m/s), the force experienced by each charge no longer depends on the current position of the other charge, but on its retarded position (t' = t − r/c). As a result, the force exerted by q2 on q1 becomes slightly stronger than the force exerted by q1 on q2. This imbalance leads to the appearance of a net force Fself acting on the whole system. This force is directed to the left and therefore tends to oppose the acceleration.
To calculate Fself, we need to compute the electric force exerted by each charge on the other, taking into account retardation effects in the propagation of the electric field. To do so, we can use the relation F = q E and compute the electric field E created by each charge using the Darwin model, which approximates field expressions to order 1/c² in the quasi-static approximation. The general expression for the field E is:
where all quantities are instantaneous (i.e. evaluated at the present time t). The first term represents the Coulomb field, and the second is a corrective term, caused by the acceleration of the charges, which precisely accounts for retardation effects. As mentioned above, the first term (though much larger in magnitude than the second) yields equal and opposite forces. The acceleration-dependent term, however, can produce forces that are not equal and opposite. Computing the net self-force Fself acting on the system gives:
F → s e l f = F → 1 → 2 + F → 2 → 1 = − 2 q 1 q 2 a → 4 π ε 0 c 2 r {\displaystyle {\vec {F}}_{\mathrm {self} }={\vec {F}}_{1\to 2}+{\vec {F}}_{2\to 1}=-{\frac {2q_{1}q_{2}\,{\vec {a}}}{4\pi \varepsilon _{0}c^{2}r}}}
The resulting force is proportional to a → {\displaystyle {\vec {a}}} and increases the system's resistance to acceleration. It can therefore be considered (by definition) as increasing the mass of the system, with the mass increment Δm given (using Newton's second law) by:
Calling this mass increment the "interaction electromagnetic mass" of the two charges, one obtains:
m e l e c = F s e l f a = 2 q 1 q 2 4 π ε 0 c 2 r {\displaystyle m_{\mathrm {elec} }={\frac {F_{\mathrm {self} }}{a}}={\frac {2q_{1}q_{2}}{4\pi \varepsilon _{0}c^{2}r}}}
Note that if the charges q1 and q2 have the same sign, the electromagnetic mass is positive (referred to as a mass excess), whereas if the charges have opposite signs, the electromagnetic mass is negative (referred to as a mass defect). Denoting by m1 the mass of charge q1, m2 the mass of charge q2, and Mtot the total mass of the system {q1+q2}, one finally has:
M t o t = m 1 + m 2 + 2 q 1 q 2 4 π ε 0 c 2 r {\displaystyle M_{tot}=m_{1}+m_{2}+{\frac {2q_{1}q_{2}}{4\pi \varepsilon _{0}c^{2}r}}}
Conclusion: The laws of classical electromagnetism show that long-range interactions between charged particles can affect the overall mass of a system, which implies that the inertia of a body does not depend solely on the amount of matter it contains. The appearance of an interaction mass is caused by the finite speed at which fields propagate through space (c). Thus, a charged capacitor will be slightly more massive than the same uncharged capacitor, even though both consist of the exact same number of particles and no matter has been exchanged with the outside. At the end of the 19th century, some physicists went so far as to claim that "mechanical" mass did not exist, and that 100% of the mass of bodies was of electromagnetic origin. However, such a hypothesis requires treating the electron and other elementary particles as spherical charge distributions, which cannot be assumed without detailed knowledge of the internal structure of these particles. Moreover, it is now known that other interactions in nature, such as gravitation and the strong nuclear interaction, can also affect the mass of bodies (nuclear mass defects are a clear illustration of this).
Transverse orientation
Considering now the situation in which the acceleration a → {\displaystyle {\vec {a}}} of the two-charge system is orthogonal to the charge axis (θ = 90°), the self-interaction force is:
F s e l f = q 1 q 2 a 4 π ε 0 c 2 r {\displaystyle F_{\mathrm {self} }={\frac {q_{1}q_{2}a}{4\pi \varepsilon _{0}c^{2}r}}}
which implies:
m e l e c = F s e l f a = q 1 q 2 4 π ε 0 c 2 r {\displaystyle m_{\mathrm {elec} }={\frac {F_{\mathrm {self} }}{a}}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}c^{2}r}}}
The interaction mass obtained is half that found in the longitudinal case.
To qualitatively understand the emergence of this self-interaction force in the transverse configuration, one can work in the accelerated frame of the two charges, where the electrostatic field lines appear curved, much like the trajectory of a tennis ball in a gravitational field. Each charge will then experience a slight electric force component opposing the acceleration, with Ex ~ Ey·(ar/c²) (following the equivalence principle of general relativity, electrostatic field lines are similarly curved for a charge at rest in a gravitational field). If the charges have opposite signs, the self-interaction force is directed along a → {\displaystyle {\vec {a}}} .
Arbitrary orientation If the charge axis makes an arbitrary angle θ with the acceleration vector a → {\displaystyle {\vec {a}}} , the interaction electromagnetic mass of the two charges is:
m e l e c = q 1 q 2 4 π ε 0 c 2 r ( 1 + cos 2 θ ) {\displaystyle m_{\mathrm {elec} }={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}c^{2}r}}\left(1+\cos ^{2}\theta \right)}
where the factor (1 + cos² θ) ranges between 1 and 2.
Appearance of a force orthogonal to the acceleration
When θ is neither 0° nor 90°, a self-interaction force perpendicular to the acceleration ( F ⊥ {\displaystyle F_{\perp }} ) also appears, in addition to the component along the acceleration:
F ⊥ = q 1 q 2 a 4 π ε 0 c 2 r cos ( θ ) sin ( θ ) {\displaystyle F_{\perp }={\frac {q_{1}q_{2}a}{4\pi \varepsilon _{0}c^{2}r}}\cos(\theta )\sin(\theta )}
However, only F ∥ {\displaystyle F_{\parallel }} can be considered as contributing to the inertia of the system. For reference:
F ∥ = q 1 q 2 a 4 π ε 0 c 2 r ( 1 + cos 2 θ ) {\displaystyle F_{\parallel }={\frac {q_{1}q_{2}a}{4\pi \varepsilon _{0}c^{2}r}}\left(1+\cos ^{2}\theta \right)}
F ⊥ {\displaystyle F_{\perp }} is maximized when θ = 45°, as can be seen from the trigonometric identity:
cos ( θ ) sin ( θ ) = 1 2 sin ( 2 θ ) {\displaystyle \cos(\theta )\sin(\theta )={\frac {1}{2}}\sin(2\theta )}
since sin(2θ) is maximized at θ = 45°.
Examples
Mass defect of the hydrogen atom
A hydrogen atom consists of two particles with opposite electric charges (a proton and an electron) in interaction. The respective masses of an isolated proton and electron are:
m p = 1 . 672621924 × 10 − 27 k g {\displaystyle m_{p}=1{.}672621924\times 10^{-27}\,\mathrm {kg} }
m e = 9 . 109383702 × 10 − 31 k g {\displaystyle m_{e}=9{.}109383702\times 10^{-31}\,\mathrm {kg} }
Their sum is:
m p + m e = 1 . 673532862 × 10 − 27 k g {\displaystyle m_{p}+m_{e}=1{.}673532862\times 10^{-27}\,\mathrm {kg} }
The experimentally measured mass of a hydrogen atom is:
m H = 1 . 673532838 × 10 − 27 k g {\displaystyle m_{H}=1{.}673532838\times 10^{-27}\,\mathrm {kg} }
One finds:
m H < m p + m e {\displaystyle m_{H}<m_{p}+m_{e}}
The difference between the two masses (the mass defect) is:
Δ m = 2 . 4 × 10 − 35 k g {\displaystyle \Delta m=2{.}4\times 10^{-35}\,\mathrm {kg} }
This mass defect is precisely the (negative) interaction electromagnetic mass between the proton and the electron. It is approximately 100 million times smaller than the total mass of the hydrogen atom.
Electromagnetic mass of a charged sphere
Consider a sphere of radius R.
Uniformly volume-charged sphere If the sphere carries a uniform volume charge density ρ throughout its volume, its electromagnetic mass can be calculated by integrating the mutual interaction forces between all infinitesimal charge pairs. The electromagnetic mass of such a charged sphere is:
M e l e c = 16 π ρ 2 R 5 45 ε 0 {\displaystyle M_{\mathrm {elec} }={\frac {16\pi \,\rho ^{2}R^{5}}{45\varepsilon _{0}}}}
This result can also be expressed in terms of the total charge Q carried by the sphere:
M e l e c = Q 2 5 π ε 0 R {\displaystyle M_{\mathrm {elec} }={\frac {Q^{2}}{5\pi \varepsilon _{0}R}}}
Uniformly surface-charged sphere A spherical shell with a uniform surface charge density σ {\displaystyle \sigma } has potential energy from the mutual repulsion in the shell. When this energy is converted to mass the result is:
M e l e c e n e r g y = Q 2 8 π ε 0 c 2 R . {\displaystyle M_{\mathrm {elec} }^{\mathrm {energy} }={\frac {Q^{2}}{8\pi \varepsilon _{0}c^{2}R}}.}
If the shell is set into motion, the momentum will imply a contribution to mass of
M e l e c m o m = 4 3 Q 2 8 π ε 0 c 2 R . {\displaystyle M_{\mathrm {elec} }^{\mathrm {mom} }={\frac {4}{3}}{\frac {Q^{2}}{8\pi \varepsilon _{0}c^{2}R}}.}
The extra 4/3 factor is attributed to Poincare stress: some force must exist to balance the mutual repulsion of the hypothetical shell and that force alters the mass.
Relation to other physical quantities
Relation between electromagnetic mass and field momentum Besides the relation m = F/a, other methods exist for computing the electromagnetic mass of a system: one of them consists of computing the momentum p → {\displaystyle {\vec {p}}} of the electromagnetic field for an arbitrary velocity v → {\displaystyle {\vec {v}}} , then using m = p/v. The electromagnetic field momentum is given by:
For two interacting bodies 1 and 2, one has:
E → = E → 1 + E → 2 {\displaystyle {\vec {E}}={\vec {E}}_{1}+{\vec {E}}_{2}} and B → = B → 1 + B → 2 {\displaystyle {\vec {B}}={\vec {B}}_{1}+{\vec {B}}_{2}}
which gives:
p → = ε 0 ∭ ( ( E → 1 × B → 1 ) + ( E → 2 × B → 2 ) + ( E → 1 × B → 2 ) + ( E → 2 × B → 1 ) ) d V {\displaystyle {\vec {p}}=\varepsilon _{0}\iiint (({\vec {E}}_{1}\times {\vec {B}}_{1})+({\vec {E}}_{2}\times {\vec {B}}_{2})+({\vec {E}}_{1}\times {\vec {B}}_{2})+({\vec {E}}_{2}\times {\vec {B}}_{1}))\,dV}
Among the four terms, the first two correspond to the individual electromagnetic masses of bodies 1 and 2. To compute only the interaction electromagnetic mass between 1 and 2, only the last two terms need to be considered. This method yields exactly the same results as the m = F/a method in all situations (both methods follow from Maxwell's equations and the Lorentz force).
Relation to magnetic energy A third method consists of computing, for a given velocity v → {\displaystyle {\vec {v}}} , the energy Um stored in the magnetic field of the system, then using m = 2Um /v². Indeed, a moving charged body accumulates and stores energy (in magnetic form) in the same way that a moving massive body accumulates kinetic energy (mass is related to kinetic energy by the relation m = 2Ec /v²). The magnetic energy Um accumulated by a moving charged body is, according to Poynting's theorem:
Since the magnetic field B generated by the body is, according to the Biot–Savart law, proportional to its velocity, the square of the magnetic field is therefore proportional to the square of the velocity, which allows a quantity homogeneous to a mass to emerge (m = 2Um/v²). For two interacting bound bodies 1 and 2, one has:
B → = B → 1 + B → 2 {\displaystyle {\vec {B}}={\vec {B}}_{1}+{\vec {B}}_{2}}
which implies:
B 2 = B 1 2 + B 2 2 + 2 B → 1 ⋅ B → 2 {\displaystyle B^{2}=B_{1}^{2}+B_{2}^{2}+2\,{\vec {B}}_{1}\cdot {\vec {B}}_{2}}
Among the three terms obtained, the first two correspond to the individual electromagnetic masses of bodies 1 and 2. If one wishes to compute only the interaction electromagnetic mass between 1 and 2, only the last term needs to be considered. In this case, one finds:
m e l e c = 2 μ 0 v 2 ∭ ( B → 1 ⋅ B → 2 ) d V {\displaystyle m_{\mathrm {elec} }={\frac {2}{\mu _{0}v^{2}}}\iiint \left({\vec {B}}_{1}\cdot {\vec {B}}_{2}\right)\,dV}
Relation to electrostatic potential energy and E=mc² The electrostatic interaction potential energy of two point charges q1 and q2 is given by:
From the earlier results for the interaction electromagnetic mass, it follows that for a system of two charges aligned along an axis parallel to the acceleration vector a → {\displaystyle {\vec {a}}} (θ = 0°):
m e l e c = 2 E p c 2 {\displaystyle m_{\mathrm {elec} }={\frac {2E_{\mathrm {p} }}{c^{2}}}}
If the charge axis is orthogonal to the acceleration vector a → {\displaystyle {\vec {a}}} (θ = 90°):
m e l e c = E p c 2 {\displaystyle m_{\mathrm {elec} }={\frac {E_{\mathrm {p} }}{c^{2}}}}
The transverse case is therefore consistent with Albert Einstein's general relation between mass and energy (E=mc²), while the longitudinal case differs from this formula by a factor of 2. It therefore appears that the relation m = E/c² yields results for the electromagnetic mass that sometimes diverge from those of the three other calculation methods (m = F/a, m = p/v and m = 2Um /v²).
The 4/3 problem For a spherical charge distribution (whether the charge is distributed on the surface of the sphere or throughout its volume), one always finds:
M e l e c = 4 3 E p c 2 {\displaystyle M_{\mathrm {elec} }={\frac {4}{3}}{\frac {E_{p}}{c^{2}}}}
This discrepancy between the results of the two approaches (E=mc² and electromagnetic theory) has generated much debate since the early 20th century. This is known as the 4/3 problem.
Poincaré stresses 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 m e m = 4 3 E e m / c 2 {\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 m e s = E e m / c 2 {\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 E p {\displaystyle E_{\mathrm {p} }} of the Poincaré stresses to E e m