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 James Clerk Maxwell 
 1831-1879 Maxwell

Final Answers
© 2000-2005 Gérard P. Michon, Ph.D.

On Maxwell's Equations

The aim of exact science is to reduce the
problems of nature to the  James Clerk Maxwell 
 1831-1879 determination
 of quantities by operations with numbers
.
James Clerk Maxwell (1831-1879) 
On Faraday's Lines of Force (1856) 

Related articles on this site:

Related Links (Outside this Site)

Faraday's Law & Lenz's Rule, by Carl R. (Rod) Nave, Georgia State U.
History of Classical Electromagnetism, by  Jeff Biggus.
The Greatest Equations Ever, by  Robert P. Crease  (Physics World, 2004).
Ampère, Gauss & Weber (21st Century Science & Technology Magazine)
Integral and Differential Forms of  Maxwells Equations.
Retarded  and  advanced  potentials,  by Richard Fitzpatrick.
Heaviside-Lorentz Units  by J. B. Calvert.
The Theory of the Electron (H. A. Lorentz, 1892)  by Fritz Rohrlich (1962).
A Gallery of Electromagnetic Personalities  by Professor Taylor.
Self-Force & Radiation Reaction  by  Luca Bombelli, University of Mississippi
 
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Electromagnetism

Note :   The modern presentation of electromagnetism incorporates two clarifications which came only many years after Maxwell's equations were published (1864): 

  • Vectorial notations and differential operators are used, which were developed by Oliver Heaviside (1850-1925) after 1880. 
  • Electromagnetic quantities come in a single "flavor" and are all expressed in units of a coherent system, as discussed next.  This system is preferably the MKSA system (Giorgi's units) which is the basis for all modern SI units:  ampere (A), volt (V), coulomb (C), volt (V), tesla (T), farad (F), henry (H), weber (Wb), etc.

Only SI electrical units are used here  (Giorgi's MKSA system).


(2005-07-22)   The  Former  Problem with Electromagnetic Units

A science which hesitates to forget its founders is lost.
Alfred North Whitehead  (1861-1947) 

This section is of historical interest only, you are advised to  skip it  if you were blessed with an education entirely grounded on Giorgi's electrodynamic units (SI units based on the MKSA system).

The first consistent system of  mechanical  units was the meter-gram-second system advocated by Carl Friedrich Gauss in 1832.  It was used by Gauss and Weber (c.1850) in the first definitions of electromagnetic units in absolute terms.

However, the term  Gaussian system  now refers to a particular mix of electrical C.G.S. units (discussed below) once dominant in theoretical investigations.

James Clerk Maxwell himself was instrumental in bringing about the  cgs system  in 1874  (centimeter-gram-second).  Two sets of electrical units were made part of the system.  An enduring confusion results from the fact that the quantities measured by these different units have different definitions  (in modern terms, for example, the magnetic quantity now denoted B could be either B or cB).  Following Maxwells own vocabulary, it's customary to speak of either  electrostatic units  (esu)  or  electromagnetic units  (emu).

Originally, none of the C.G.S. electromagnetic units had specific names.  On August 25th, 1900, the International Electrical Congress adopted two names :

  • Gauss  for the C.G.S. unit of magnetic field:   1 G  =  10 - T.
  • Maxwell  for the C.G.S. unit of magnetic flux:   1 Mx  =  10 - Wb.

The  maxwell, still known as a  line of force, is called  abweber  (abWb)  using the later naming of CGS electrical units after their MKSA counterparts.  Likewise, the  gauss  (1 maxwell per square centimeter)  is also called  abtesla  (abT).  For  electrostatic  CGS units  (esu)  the prefix  stat-  is used instead...

Electrodynamic units are now based on a proper independent electrical unit, as advocated by the Italian engineer Giovanni Giorgi (1871-1950) in 1901:  The addition of the  ampère  to the MKS system has turned it into a consistent 4-dimensional system (MKSA) which is the foundation for modern SI units.

 Come back later, we're
 still working on this one...

Scholars from bygone days should be credited for accomplishing so much  in spite of  such confusing systems.


(2005-07-15)   The Lorentz Force
The Lorentz force on a test particle  defines  the electromagnetic field(s).

The expression of the Lorentz force introduced here defines dynamically the fields which are governed by Maxwell's equations, as presented further down.  Neither of these two statements is a logical consequence of the other.

We offer no apology for our resolutely anachronistic presentation:  The concept of an electric field is due to Michael Faraday (1791-1867) while the modern expression of the force exerted by an electromagnetic field on a moving electric charge is due to H.A. Lorentz (1853-1928).

In  electrostatics,  the electric field  E  present at the location of a particle of charge  q  summarizes the influence of all other electric charges, by stating that the particle is submitted to an electrostatic force equal to q E.  This defines  E.

This  concept  may be extended to  magnetostatics  for a moving test particle.  More generally, the electromagnetic fields need not be constant in the following expression of the force acting on a particle of charge  q  moving at velocity  v.

The Lorentz Force
F   =   q  ( E + v´B )

The average force exerted per unit of volume may thus be expressed in terms of the  density of charge  r  and the  density of current  j.

Density of Force
dF / dV    =    r E   +   j ´ B


(2005-07-18)   Electrostatics:   On the electric field from static charges.
Coulomb's  inverse square law  translates into the local differential property of the field expressed by Gauss, namely:   div E = r/eo

 Charles Augustin de Coulomb 
 1736-1806 Coulomb The SI unit of electric charge is named after the French military engineer  Charles Augustin de Coulomb (1736-1806).  Using a  torsion balance,  Coulomb discovered, in 1785, that the electrostatic force between two charged particles is proportional to each charge, and inversely proportional to the square of the distance between them.  In modern termsCoulomb's Law  reads:

Electrostatic Force  between Two Charged Particles
|| F ||    =     | q  q' |
vinculum
4peo r 2

The  direction  of the electrostatic force is on the line joining the two charges.  The force is  repulsive  between charges of the same sign  (both negative or both positive).  It's  attractive  between charges of  unlike  signs.

In the language of fields introduced above, all of the above is summarized by the following expression, which gives the  electrostatic  field  E  produced at position  r  by a motionless particle of charge  q  located at the origin:

Electrostatic Field of a Point Charge at the Origin
E     =     q  r
vinculum
4peo r 3

Since  r / r 3  is the  opposite  of the gradient of  1/r,  we may rewrite this as :

E    =    - grad f   where     f    =     q 
vinculum
4peo r

The additivity of forces means that the contributions to the local field  E  of many remote charges are additive too.  The  electrostatic potential  f  we just introduced may thus be computed additively as well.  This leads to the following formula, which reduces the computation of a  three-dimensional  electrostatic field to the integration of a  scalar  over any  static  distribution of charges:

The Electrostatic Field  E  and Scalar Potential  f
E    =    - grad f   where     f(r)    =    òòò    r (s)   d3s 
vinculum
4peo || r - s ||

The above  static  expression of  E  would have to be completed with a dynamic quantity  (namely  A/t,  as discussed below)  in the  nonstatic  case governed by the full set of Maxwell's equations.  Also, the  dynamical  scalar potential  f  involves a more delicate integration than the above one.

 Carl F. Gauss 
 1777-1855 Coulomb In 1832, Gauss  bypassed both dynamical caveats with a  local  differential expression,  also valid in  electrodynamics :

div E     =     r 
vinculum
eo

  The reader may want to establish this relation with elementary methods. 
HINT:  One way to do so is to start by checking that the above electric field of a point charge has a vanishing divergence at every point outside of the source.  On the other hand, the integral of the divergence in any sphere centered on the source  (regardless of radius)  is the flux of  E  through the surface of such a sphere, which is readily seen to be equal to   q /eo  QED

Gauss's Theorem of Electrostatics  (1832)

 Boundary of a Volume

In electrostatics, we call  Gauss's Theorem  the integral equivalent of the above differential relation, namely:

Q / eo     =     òòòV   ( r / eo )  dV     =     òòS   E . dS

This states that the  outward flux  of the electric field  E  through a surface bounding any given volume is equal to the electric charge  Q  contained in that volume, divided by the permittivity  eo.

The next section features a typical example of the use of  Gauss's Theorem.

Another interesting consequence is that the field  outside  any distribution of charge with spherical symmetry has the same expression as the field which would be produced if the entire electrical charge was concentrated at the origin.

This property of inverse square laws was first discovered with elementary methods in the context of gravitation, by working out the newtonian field  outside  an homogeneous spherical shell  (incidentally, the field  inside  such a shell is zero).  This means that, in the main, a planet behaves like a point of the same mass located at its center.


(2005-07-20)   Electric Capacity  [ electrostatics, or  low  frequency ]
The static charges on conductors are proportional to their potentials.

Consider an horizontal foil carrying a  supercifial charge  of  s  C/m2.  Let's limit ourselves to points that are close enough to [the center of] the plate to make it look practically infinite.  Symmetries imply that the field is vertical  (the electrical flux through any vertical surface vanishes)  and that its value depends only on the altitude  z  above the plate  (also, if it's E at altitude z, then it's -E at altitude -z).

Let's apply  Gauss's theorem  to a vertical cylinder whose horizontal bases are above and below the foil, each having area  S.  This box contains a charge   sS  and the electrical flux out of it is  2 E S.  Therefore, we obtain for  E  a  constant  value, which does  not  depend on the distance  z  to the plate:  E = ½ s/eo.

Capacitor consisting of two parallel plates :

For two parallel foils with opposite charges, the situation is the  superposition  of two distributions of the type we just discussed:  This means an electric field which vanishes outside of the plates, but has twice the above value between them.

Assuming a small enough distance  d  between two plates of a large surface area  S,  the above analysis is supposed to be good enough for most points between the plates  (so what happens close to the borders is negligible).  The whole thing is called a  capacitor  and the following quantity is its  electric capacity.

Capacity of Two Parallel Plates
C     =     eo S
vinculum
 d 

Because  E  =  s / eo  =  q / Seo  =  -¶f/¶z ,   the difference of the potentials  f  of the two plates is  qd / Seo  =  q/C.  In other words:

Charge on a Capacitor's Plate
q   =   C U

This is a general relation.  In a static  (or nearly static)  situation, the potential is the same throughout the conductive material of each plate.  The proportionality between the field and its sources imply that the charge q on one plate is proportional to the difference of potential between the two plates.  We  define  the capacity as the relevant coefficient of proportionality.

Permittivity of Dielectric Materials :

The above holds only if the space between the capacitor's plates is empty  (air being a fairly good approximation for emptiness).  In practice, a dielectric material may be used instead, which behaves nearly as the vacuum would if it had a different permittivity.  This turns the above formula into the following one.  In electrodynamics, the permittivity  e  may depend  a lot  on frequency.

C     =     e S
vinculum
d

capacity  is  e  times a geometrical factor, homogeneous to a  length.


(2005-07-19)   Faraday's Law of Electromagnetic Induction
A varying magnetic flux induces an electric circulation around the border.

 Michael Faraday 
 1791-1867 Michael Faraday (1791-1867) was the son of a blacksmith, and a bookbinder by trade...  In 1810, he started attending the lectures that Humphry Davy (1778-1829) had been giving at the Royal Institution of London since 1801.  In December 1811, Faraday became an assistant of Davy, whom he would eventually surpass in knowledge and influence.  Faraday was elected to the Royal Society in 1824, in spite of the jealous opposition of Sir Humphry Davy who was its president  (from 1820 to 1827).

Arguably, the greatest of Faraday's many scientific contributions was the  Law of Induction  which he formulated in 1831.  By explaining the 1821 observation of Oersted in terms of what we now called the magnetic  field, Faraday did much more than invent the electric motor, he opened entirely new vistas for physics.  He proposed that light itself was an electromagnetic phenomenon.  He lived to be proven right mathematically by his younger friend, James Clerk Maxwell.

 Heinrich Lenz 
 1804-1865
Heinrich Lenz 
Heinrich Lenz (1804-1865).
Lenz's Law (1833).

The magnetic flux...
F = B . S
dF = dB . S  +  B . dS
First term = Magnetic Induction.   Second Term = Lorentz Force.

 Come back later, we're
 still working on this one...

(2005-07-18)   On the History of Maxwell's Equations
The basic laws of electricity and magnetism.

Gauss' Electric Law = Coulomb's Law
Gauss's Magnetic Law.
Faraday's Law of Induction.
Ampère's Law.

 Come back later, we're
 still working on this one...

(2005-07-09)   Maxwell's Equations Unify Electricity and Magnetism
They predicted electromagnetic waves before Hertz demonstrated them.

Maxwell's equations   govern the electromagnetic quantities defined above :

  • The  electric field  E
  • The  magnetic induction  B
  • The  density of electric charge  r
  • The  density of electric current  j

Maxwell's Equations  (1864)
rot E  +      B   =   0   div E  =   r 
vinculum vinculum
t eo
rot B  -   1  E   =   mo j div B  =   0
vinculum vinculum
c2  t

The three electromagnetic  constants  involved are tied by one equation:

emc 2   =   1

A direct consequence of Maxwell's equations is the following relation, which expresses the  conservation of electric charge  (HINT:  div rot B  vanishes...  The conclusion holds  if and only if  the 3 constants are related as advertised.)

Electric Charge is Conserved
div j  +    r   =   0
vinculum
 t

Using the identity   rot rot V = grad div V - DV   when   r = 0   and   j = 0, we obtain the following equality for  any  electromagnetic component  y.

   2 y     =     Dy
Vinculum Vinculum
c 2 t 2

This  wave equation  shows that the electromagnetic field propagates at  celerity  c  in a vacuum.  This led Maxwell to propose an  electromagnetic theory of light.  The propagation of electromagnetic waves  (radio waves)  was demonstrated experimentally in 1888, by  Heinrich Rudolf Hertz  (1857-1894).


(2005-07-15)   Electromagnetic Planar Waves   (Progressive Waves)
The simplest solutions to  Maxwell's equations, far from all sources.

In the absence of electromagnetic sources  ( r = 0,  j = 0 )  we may look for electromagnetic fields whose values do not depend on the y and z coordinates.  A solution of this type is called a  progressive planar wave  and it's best established directly from the above  equations of Maxwell, without resorting to the electromagnetic potentials introduced below.

 Come back later, we're
 still working on this one...

(2005-07-15)   Electromagnetic Energy  & Poynting Vector
The Lorentz force transfers energy from the field to the charge carriers.

The power  F.v  of the Lorentz force is  q E.v.  Thus, the power received by the electric charges per unit of volume is  E.j.  The charge carriers may then convert the power so received from the local electromagnetic field into other form of energy  (including the kinetic energy of particles). 

E.j   may be expressed as follows, in terms of the electromagnetic fields alone.  HINT:  Express  j  in terms of the fields with Maxwell's equations, then use the identity   E.rot B  =  B.rot E - div E´B   and the relation   rot E = B/t  QED

Electromagnetic Balance of Energy Density   (Poynting Theorem, 1884)
div  (   E ´ B   )     +       (   eo E 2  +  B 2 / mo   )     =     - E . j
vinculum vinculum vinculum
mo  t 2

This relation is due to a former student of Maxwell,  John Henry Poynting  (1852-1914).  The vector  S = E´B / mo  is now called the  Poynting vector.

The right-hand side of of the above equation is the  opposite  of the power delivered by the field to the sources, per unit of volume.  It's therefore the density of the power released by the sources to the field.  The left-hand side is readily interpreted as evidence for an energy carried by the electromagnetic field:

Electromagnetic Energy Density
1/2  eo  (  E 2  +  c2 B 2  )

In a given volume, this energy is either delivered directly by local sources or it comes from radiation through its surface, as the flux of the  Poynting vector.


(2005-07-13)   Electromagnetic Potentials  &  Lorenz Gauge
Devised by Ludwig Lorenz in 1867  [when H.A. Lorentz was only 14].

Since Maxwell's equations assert that the divergence of  B  vanishes, there is necessarily a  vector potiental  A  of which  B  is the rotational (or curl).

B   =    rot A

This implies that   rot [ E + A/t ] = 0   which proves that the bracket is the gradient of a  scalar potential, called   -f  for compatibility with electrostatics.

E   =   - grad f - ¶A/t

These two equations  do not  uniquely determine the potentials, as the same fields are obtained with the following changes in the potentials, for any scalar field   y.

A   ¬   A  +  grad y               f   ¬   f  -  ¶y/¶t

This leeway can be used to make sure the following equation is satisfied, as proposed by Ludwig Lorenz in 1867.  (Watch the spelling...  There's no "t".)

The Lorenz Gauge  (1867)
div A  +   1     f    =  0
vinculum vinculum
c2  t

The  Lorenz Gauge  still allows the above type of leeway, but  only  for an arbitrary field  y  propagating freely at celerity  c, according to the equation:

   2 y     =     Dy
Vinculum Vinculum
c 2 t 2

The two Maxwell equations which don't involve electromagnetic sources are equivalent to the above definitions of  E  and B  in terms of electromagnetic potentials.  Using the Lorenz Gauge, the other two equations boil down to the following relations between the electromagnetic sources and the potentials:

D'Alembert's  Equations
Df   -   1     f     =       r
vinculum vinculum minus vinculum
c2  t eo
DA   -   1  A     =   - mo j
vinculum vinculum
c2  t

Without the  Lorenz Gauge,  more complicated relations would hold:

Df   -   1     f =     r         ( div A  +   1     f   )
vinculum vinculum minus vinculum minus vinculum vinculum vinculum
c2  t eo  t c2  t
DA   -   1  A = - mj  +  grad   ( div A  +   1     f   )
vinculum vinculum vinculum vinculum
c2  t c2  t

Although it was once a mere mathematical convenience, the  Lorenz gauge  is now considered fundamental, because quantum theory assigns a  physical  significance to electromagnetic potentials.  In particular, the following definition of the so-called  canonical momentum  makes the  Lorenz gauge  mandatory:

Canonical momentum of a particle of
mass
  m,  charge  q   and velocity  v
p    =    m v  +  q A

The Aharonov-Bohm Effect


(2005-07-15)   Retarded and Advanced Potentials
General solutions of  Maxwell's equations  obeying the  Lorenz gauge.

As shown below, the following potentials satisfy the Lorenz gauge 

Electrodynamic Retarded Potentials   A-  and  f-
f-(t,r) = òòò    r ( t - ||r-s|| / c  ,  s )   d3s 
vinculum
4peo || r - s ||
A-(t,r) = òòò    mo j ( t - ||r-s|| / c  ,  s )   d3s 
vinculum
4p || r - s ||

This is similar to the expressions obtained in the static cases  (electrostatics, magnetostatics)  except  that the fields we observe  here and now  depend on a prior state of the sources.  The influence of the sources is delayed by the time it would take the "news" of their changing state to travel at speed c.

There's also another solution  (the so-called  advanced  potentials  A+ and f+ )  which is formally obtained from the above by changing the sign of c,  or  equivalently  by reversing the  arrow of time.  This is just like what we've already encountered in the case of planar waves, with two possible directions of travel.  However, the physical interpretation is not nearly as easy now that we're dealing with some causality relationship between the field and its "sources". 

Advanced potentials  make the situation here and now  (potentials and/or fields)  depend on the  future  state of remote "sources".  Such a thing may be summarily dismissed as "unphysical" but this fails to make the issue go away.  Indeed, quantum treatments of electromagnetic fields  (photons in  Quantum Field Theory )  imply that a field can create some of its sources in the form of charged particle-antiparticle pairs.  What seems to be lacking is the  coherence  of such creations because of statistical and/or thermodynamical considerations  (which feature a pronounced arrow of time).  I don't understand this.  Nobody does...

What's clear, however, is that the distinction between past and future vanishes in  stationary  cases.  This makes  advanced potentials  relevant and/or necessary, without the need for mind-boggling philosophical considerations.


Here's the outline of a proof that the  retarded potentials  satisfy the  Lorenz gauge, as advertised...  (The same is true of the advanced potentials.)

Consider the integrands for a fixed position of the source  (that's what  s  is). 

 Come back later, we're
 still working on this one...

(2005-08-21)   Electrodynamic Fields
A general expression derived from the above retarded potentials.

Letting  R  be  || r - s ||  the retarded potentials yield the following fields, where  r  and  j  stand, respectively, for   r ( t - R / c  ,  s )   and   j ( t - R / c  ,  s ).

Electrodynamic fields obtained from retarded potentials :
E(t,r)=  1    òòò  [   r ( r - s )   +   ( ¶ r / t ) ( r - s )   -   j / t   ]   d3s
vinculum
4peo  R 3 c R 2 c 2 R
B(t,r)= mo   òòò  [   j ´ ( r - s )   +   ( j / t ) ´ ( r - s )     ]   d3s
vinculum
4p  R 3 c R 2

The first term of each expression corresponds to electrostatics & magnetostatics.


(2005-07-15)   Electric Moment  &  Magnetic Moment
The electrodynamic fields of dipoles.

The following expressions could be obtained from the general expressions of electrodynamic potentials and/or fields in the limit of dipolar distributions.

However, the author fondly remembers establishing both sets of dipolar formulas  (as a student, in June of 1975 or 1976)  by proving that, if there are no sources at a nonzero distance from the origin, linear superpositions of these two are the only "spherical and dipolar" solutions of Maxwell's equations.  Loosely speaking, this is to say that there's no other way to build solutions of Maxwell's equations where the value of each field component at position  r  is a sum of products of  k(r)r  by a vector  Z(t-r/c)  or by one of its derivatives...

The two types of dynamic solutions that emerge from such an analysis are readily identified from the respective  static  parts of the electric and magnetic dipolar fields.  (These well-known static fields are obtained as the limiting cases of simple distributions  [two point charges, or a current loop]  whose moments are kept constant as their sizes tend to zero.)

My long-forgotten motivation was to use such solutions as rigorous building blocks for dealing with interference using Huygens' principle.

The potentials listed below both satisfy the Lorenz gauge.

Electric Dipole

The electric dipolar moment  P  of a confined charge distribution is defined as:

P     =       òòò   r r dV   -   r òòò   r dV

The second term is zero if there's not net electric charge, in which case the value of the first term does not depend on the origin chosen for positions.  The dipolar moment of a neutral distribution of point charges is  P = å qri.

Electrodynamic Field of an Electric Dipole at the Origin     P = P(t-r/c)   u = r/r
ì
ï
í
ï
î
f=  1     [       u . P       +       u . P'     ]
vinculum vinculum vinculum
4peo r r c
A= mo    [       P'     ]
vinculum
4p r
 
ì
ï
í
ï
î
E=  1     [   3 (u.P) u - P   +   3 (u.P' ) u - P'   +   u ´ (u ´ P'' )   ]
vinculum
4peo r  r 2 c r  c 2
B= - mo    [         u ´ P'       +       u ´ P''       ]
vinculum vinculum vinculum
4p r

The counterpart of the above for magnetic dipoles is discussed below.

Magnetic Dipole

The magnetic dipolar moment  M  of a confined current distribution is:

M     =       ½ òòò   r ´ j  dV   -   ½  r ´ òòò   j dV

The second term is zero for confined currents, in which case the value of the first term does not depend on the origin chosen for positions.  The dipolar moment for a current I flowing in a loop of vectorial area  S  is  M = I S.

Electrodynamic Field of a Magnetic Dipole at the Origin     M = M(t-r/c)   u = r/r
ì
ï
í
ï
î
f = 0 
 
A= - mo   [      u ´ M       +       u ´ M'     ]
vinculum vinculum vinculum
4p r r c
 
ì
ï
í
ï
î
E= mo   [        u ´ M'       +       u ´ M''       ]
vinculum vinculum vinculum
4p r
B= mo   [  3 (u.M) u - M   +   3 (u.M' ) u - M'   +   u ´ (u ´ M'' )   ]
vinculum
4p r  r 2 c r  c 2


(2005-08-11)   Radiated Energy
Accelerated charges radiate energy away.

Consider the above dipolar fields.  At a large distance, the dominant field components are those proportional to the second derivatives  P''  or  M''.

 Come back later, we're
 still working on this one...

(2005-08-09)   The Lorentz-Dirac Equation
Classical Theory of the Electron.  Strange inertia of charged particles.

The motion of an electron (point particle of charge q) submitted to a force  F  has been described in terms of the following 4-dimensional equation, where (primed) derivatives of the position  R  are with respect to the particle's  proper time  t  [ which may be defined via:  (c dt)2 = (c dt)2-(dx)2-(dy)2-(dz)].

Lorentz-Dirac Equation   (1938)
m R''    =    F  +   mo q 2   [  R'''  +   | R' > <R' |  R''' ]
vinculum vinculum
6p c c 2
The Abraham-Lorentz  equation is the non-relativistic version of this  (using "absolute" time and retaining only the first term of the bracket).

| R' ><R' |  is a square tensor  (the product of the 4D velocity and its dual).  The bracketed sum is only relevant for a point particle of nonzero charge.  Its nature has been highly controversial since 1892, when H.A. Lorentz first proposed a  Theory of the Electron  derived microscopically from Maxwell's equations and from the expression of the electromagnetic force now named after him.  Lorentz would only consider the electromagnetic part of the  rest mass  m  (i.e., 3m/4).  In 1938, Paul Dirac derived the above  covariantly, for the total mass  m.

Physically, the initial value of the acceleration (R'' ) in this third-order equation cannot be freely chosen  (so the overall constraints are comparable to those of an ordinary newtonian equation).  Almost all  mathematical  solutions are unphysical ones, which are dubbed  self-accelerating  or  runaway  because they would make the particle's energy grow indefinitely, even if no force was applied.

However, there could be  more than one  initial value of the acceleration which makes physical sense.  The  wholly classical  Lorentz-Dirac equation thus allows a nondeterministic behavior more often associated with quantum mechanics.

The Lorentz-Dirac equation has other  weird  features, including the need for a so-called  preacceleration  which contradicts causality:  The Lorentz-Dirac equation would require an electron to anticipate any impending pulse of force...

Order Reduction for the Lorentz-Dirac Type   |   Does a Uniformly Accelerating Charge Radiate?

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