On the Notation of MAXWELL’s Field Equations

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On the Notation of MAXWELL’s

Field Equations

André Waser* Issued: 28.06.2000

Last revison: -

Maxwell’s equations are the cornerstone in electrodynamics. Despite the fact

that this equations are more than hundred years old, they still are subject to

changes in content or notation. To get an impression over the historical development

of Maxwell’s equations, the equation systems in different notations are

summarized.

Introduction

The complete set of the equations of James Clerk MAXWELL[15] are known in electrodynamics

since 1865. These have been defined for 20 field variables. Later Oliver HEAVISIDE[11] and

William GIBBS[23] have transformed this equations into the today’s most used notation with

vectors. This has not been happened without ‚background noise‘[3], then at that time many

scientists – one of them has been MAXWELL himself – was convinced, that the correct notation

for electrodynamics must be possible with quaternions[5] and not with vectors. A century later

EINSTEIN introduced Special Relativity and since then it was common to summarize MAXWELL’s

equations with four-vectors.

The search at magnetic monopoles has not been coming to an end, since DIRAC[4] introduced

a symmetric formulation of MAXWELL’s equations without using imaginary fields. But

in this case the conclusion from the Special Theory of Relativity, that the magnetic field

originates from relative motion only, can not be hold anymore.

The non-symmetry in MAXWELL’s equations of the today’s vector notation may have disturbed

many scientists intuitively, what could be the reason, that they published an extended

set of equations, which they sometime introduced for different applications. This essay summarizes

the main different notation forms of MAXWELL’s equations.

* André Waser, Birchli 35, CH-8840 Einsiedeln; andre.waser@aw-verlag.ch

Maxwell’s Equations

The Original Equations

With the knowledge of fluid mechanics MAXWELL[15] has introduced the following eight

equations to the electromagnetic fields (the right equations correspond with the original text,

the left equations correspond with today’s vector notation):

1 1

2 2

3 3

d D

p p J j

dt t

d D

q q J j

dt t t

d D

r r J j

dt t

ü ¶ ü = + ï = + ï ï ¶ ï

ï ¶ ï ¶ = + ý ® = + ý Þ = + ï ¶ ï ¶

ï ¶ ï

= + ï = + ï þ ¶ þ

J j (1.1)

3 2

1 3

2 1

dH dG A A

dy dz y z

dF dH A A

dz dx z x

dG dF A A

dx dy x y

ü ¶ ¶ ü ma = - ï m = - ï ï ¶ ¶ ï

ï ¶ ¶ ï

mb = - ý ® m = - ý Þ m =Ñ´

ï ¶ ¶ ï

mg = - ï m = - ï þ ¶ ¶ þ

H A (1.2)

3 2

1 3

2 1

d d H H

4 p 4 J

dy dz y z

d d H H

4 q 4 J

dz dx z x

d d H H

4 r 4 J

dx dy x y

g b ü ¶ ¶ ü - = p ï - = p ï ï ¶ ¶ ï

a g ï ¶ ¶ ï

- = p ý ® - = p ý Þ Ñ´ =

ï ¶ ¶ ï

b a ï ¶ ¶ ï

- = p ï - = p ï þ ¶ ¶ þ

H J (1.3)

( )

1 3 2 2 3

2 1 3 3 1

3 2 1 1 2

dy dz dF d dA d P E H v H v dt dt dt dx dt dx

dz dx dG d dA d

Q E Hv Hv

dt dt dt dy dt dy

dx dy dH d dA d

R E H v H v

dt dt dt dz dt dz

æ ö Y ü jü = mç g -b ÷ - - ï = m - - - ï è ø ï ï

æ ö Y ïï jï = mça - g ÷ - - ý ® = m - - - ý

è ø ï ï

æ ö Yï jï = mçb -a ÷ - - ï = m - - - ïè ø ïþ þ

Þ =m ´ - -Ñj

E v H

(1.4)

1 1

2 2

3 3

P k E D

Q k E D

R k E D

= ü e = ü

= ï ® e = ï Þ e = ý ý

= ï e = ï þ þ

E D (1.5)

1 1

2 2

3 3

P p E j

Q q E j

R r E j

= -z ü s = ü

= -z ï ® s = ï Þ s = ý ý

= -z ï s = ï þ þ

E j (1.6)

1 2 3 d d d D D D

e 0 0

dx dy dz x y z

f g h ¶ ¶ ¶

+ + + = ® r+ + + = Þ - r =Ñ

¶ ¶ ¶

gD (1.7)

1 2 3de dp dq dr j j j

0 0

dt dx dy dz t x y z t

¶r ¶ ¶ ¶ ¶r

+ + + = ® + + + = Þ - =Ñ

¶ ¶ ¶ ¶ ¶

gj (1.8)

This original equations do not strictly correspond to today’s vector equations. The original

equations, for example, contains the vector potential A, which today usually is eliminated.

Three Maxwell equations can be found quickly in the original set, together with OHM’s

law (1.6), the FARADAY-force (1.4) and the continuity equation (1.8) for a region containing

charges.

The Original Quaternion Form of Maxwell‘s Equations

In his Treatise[16] of 1873 MAXWELL has already modified his original equations of 1865. In

addition Maxwell tried to introduce the quaternion notation by writing down his results also

in a quaternion form. However, he has never really calculated with quaternions but only uses

either the scalar or the vector part of a quaternion in his equations.

A general quaternion has a scalar (real) and a vector (imaginary) part. In the example below

‚a‘ is the scalar part and ‘ib + jc + kd’ is the vector part.

Q = a + ib + jc + kd

Here a, b, c and d are real numbers and i, j, k are the so-called HAMILTON‘ian[7] unit vectors

with the magnitude of Ö-1. They fulfill the equations

i2 = j2 = k2 = ijk = -1

and

ij = k jk = i ki = j

ij = - ji jk = - kj ki = - ik

A nice presentation about the rotation capabilities of the HAMILTON’ian unit vectors in a

three-dimensional ARGAND diagram was published by GOUGH[6].

Now MAXWELL has defined the field vectors (for example B = B1i + B2j + B3k) as quaternions

without scalar part and scalars as quaternions without vector part. In addition he

defined a quaternion operator without scalar part

1 2 3

d d d

dx dx dx

Ñ = i+ j+ k ,

which he used in his equations. Maxwell devided a single quaternion with two prefixes into a

scalar and vector. This prefixes he defined according to

S.Q = S.(a + ib + jc + kd) = a

V.Q = V.(a + ib + jc + kd) = ib + jc + kd

The original Maxwell quaternion equations are now for isotrope media (no changes except

fonts, normal letter = scalar, capital letter = quaternion without scalar):

B = V.ÑA (1.9)

E = V.vB- A& -ÑY (1.10)

F = V.vB+ eE -mÑW (1.11)

B = H + 4pM (1.12)

tot 4pJ = V.ÑH (1.13)

J = CE (1.14)

D KE

(1.15)

tot J= J + D& (1.16)

B = mH (1.17)

e = S.ÑD (1.18)

m = S.ÑM (1.19)

H= -ÑW (1.20)

Beneath the new notation, the magnetic potential field W and the magnetic mass m was

mentioned here the first time. By calculating the gradient of this magnetic potential field it is

possible to get the magnetic field (or in analogy the magnetostatic field. Maxwell has introduced

this two new field variables into the force equation (1.11).

The reader may check that the equations above identical to the previous published equations

(1.1) bis (1.7), except the continuity equation (1.8) has this time be dropped. From the

above notation it is clearly visible why the quaternion despite the deep engagement for example

of Professor Peter Guthrie TAIT[19] did not succeed, then the new introduced vector notation

of Oliver HEAVISIDE[11] and Josiah Willard GIBBS[23] was much easier to read and to use

for most applications.

It is very interesting that Maxwell‘s first formulation of a magnetic charge density and the

related discussion about the possible existence of magnetic monopoles became forgotten for

more than half a century until in 1931 Paul André Maurice DIRAC[4] again speculated about

magnetic monopoles.

Today‘s Vector Notation of MAXWELL‘s Equations

The nowadays most often used notation can be easily derived from the original equations of

1865. By inserting (1.1) in (1.3) it follows the known equation

Ñ´ = +

H j (1.21)

Equation (1.4) contains the FARADAY equation

E = v´mH ¾m¾=ko¾nsta¾nt® E = m( v´H) (1.22)

and the potential equation for the electric field

=- -Ñj

E . (1.23)

Together with the potential equation for the magnetic field (1.2) follows with applying the

rotation on both sides of (1.23)

( ) ( ) konstant

t t t

m= ¶ ¶ ¶

Ñ´ =- Ñ´ = m ¾¾¾¾® Ñ´ =m

¶ ¶ ¶

E A H E (1.24)

From (1.2) follows further with the divergence:

ÑgmH = 0 ¾m¾=ko¾nsta¾nt® mÑgH = 0 (1.25)

The six MAXWELL equations in today‘s notation are:

FARADAY‘s law

Ñ´ = +

H j (1.26)

AMPÈRE‘s law

-Ñ´ =

E (1.27)

COULOMB‘s law ÑgD = -r (1.28)

ÑgB = 0 (1.29)

0 0 r D = e E+ P = e e E = eE (1.30)

0 0 r B = m H +M = m m H = mH (1.31)

with

E: electrical field strength [V / m]

H: magnetic field strength [A / m]

D: electric displacement [As / m2]

B: magnetic Induction [Vs / m2]

j: electric current density [A / m2]

e: electric permeability [As / Vm]

m: magnetic permeability [Vs / Am]

Please note that the MAXWELL equation of today have became subset of the original equations

which in turn have got an expansion with the introduction of the magnetic induction (1.31).

Today traditionally not included in MAXWELL‘s equations are FARADAY‘s law and sometime

also OHM‘s law. Seldom the continuity equation (1.8) is even mentioned. But this

equation defines the conservation of charge:

( ) ( ) 0 0

t t t t

¶ ¶r ¶r ¶r

Ñ Ñ´ = Ñ +Ñ =- - = Þ =

¶ ¶ ¶ ¶

g H gD gj (1.32)

The electric and magnetic field strengths are interpreted as a physically existent force fields,

which are able to describe forces between electric and magnetic poles. Maxwell has – analogue

to fluid mechanics – this force fields associated with two underlying potential fields,

which are not shown anymore in the today‘s traditional vector notation. The force fields can

be derived from the potential fields as:

- = Ñj+

E (1.33)

B =Ñ´A (1.34)

with

j: electric potential field [V]

A: vector potential [Vs / m]

For a very long time scientists are convinced that the potentials do not have any physical

existence but merely are a mathematical construct. But an experiment sugested by Yakir

AHARONOV and David BOHM[1] has shown, that this is not true. There arises the question

about the causality of the fields. Many reasons point out that the potentials j and A really are

the cause of the force fields E and H.

Including the material equations (1.30) and (1.31) and with consideration of Ohm‘s law

j = sE (1.35)

with

s: specific electric conductivity [1 / Wm] = [A / Vm]

the Maxwell equations become for homogenous and isotrope conditions (e = constant,

m = constant):

Ñ´ =e +s

H E (1.36)

-Ñ´ =m

E (1.37)

eÑgE = r (1.38)

mÑgH = 0 (1.39)

Real Expansions of Maxwell‘s Equations

The HERTZ-Ansatz

Recently Thomas PHIPPS[20] has shown that Heinrich Rudolf HERTZ has suggested another

possibility to adapt Maxwell‘s equations. During Hertz life this was hardly criticized and his

proposal was vastly forgotten after his death. Usually the differentials are partial derivative

and not total derivatives as shown in the comparison (1.1) to (1.8) between MAXWELL‘s

original equations and the today‘s vector notation. Now in the equations (1.26) and (1.27)

HERTZ has substituted the partial derivatives ¶ with the total derivatives d. With this the

Maxwell equations become invariant to the GALILEI-transformation:

Ñ´ = +

H j (1.40)

-Ñ´ =

E (1.41)

what wit the entity

dt t

= + Ñ

vg becomes

Ñ´ = + Ñ +

H vg D j (1.42)

-Ñ´ = + Ñ

E vg B (1.43)

Now the question arises about the meaning of the newly introduced velocity v. HERTZ has

interpreted this velocity as the (absolute) motion of aether elements. But if v is interpreted as

relative velocity between charges, then Maxwell‘s equations are defined for the case v = 0, hat

can be interpreted that the test charge does not move in the observer‘s reference frame.

Therefore Thomas PHIPPS explains this velocity as the velocity of a test charge relative to an

observer.

Consequently in equation (1.33) the partial derivatives has to be replaced wit the total derivatives,

too.

dt t

- = Ñj+ = Ñj+ + Ñ

A A

E vg A (1.44)

The invariance of (1.40) and (1.41) against a GALILEI-transformation for the case that no

current densities j and no charges are present can easily be seen. For v = 0 (a relative to the

observer stationary charge) always MAXWELL‘s equations will be the result:

Ñ´ =

H (1.45)

-Ñ´ =

E (1.46)

For a GALILEI-transformation is r‘ = r – vt and t‘ = t; thus for v > 0 is:

x x y y z z

' '

¶ ¶ ¶ ¶ ¶ ¶

= = = ® Ñ =Ñ

¶ ¢ ¶ ¶ ¶ ¶ ¶

and

t' t

¶ ¶

= + Ñ

¶ ¶

from which for all v the equations

Ñ´ =

H (1.47)

-Ñ´ =

are valid. If the observer moves together with a test charge, this reduces again to the equations

(1.45) and (1.46). The first EINSTEIN postulate[5], that in an uniform moving system all

physical laws take its simplest form independent of the velocity, is in the example above

fulfilled. In each uniform moving reference frame the observer always measures for example

the undamped wave equation.

The DIRAC-Ansatz

The non-symmetry in MAXWELL‘s equation system always has motivated to extend this set of

equations. The most famous extension has originated form DIRAC [3], who suggested the

following extension:

e t

Ñ´ =e +

H j (1.48)

m t

-Ñ´ =m +

E j (1.49)

eÑgE = re (1.50)

m mÑgH = r (1.51)

Together with (1.51) this ansatz must lead to the postulation of magnetic monopoles, which

until today never has been (absolutely certain) detected. As a consequence of this ansatz the

force fields E and B are derived from potentials according to:

= Ñj- -Ñ´

E C (1.52)

= Ñf- -Ñ´

E A (1.53)

where f and C represent the complementary magnetic potentials. Therefore as another

consequence there must exist two different kinds of photons, which interact in different ways

with matter[14]. Also this has until today never been observed.

The HARMUTH-Ansatz

Henning HARMUTH[5] and Konstantin MEYL[17] have gone a step further and suggested new

equations, which differ to the DIRAC ansatz only in that point, that no source fields exists

anymore. Harmuth has used this proposition to solve the problem of propagation of electromagnetic

impulses in lossy media (impulses in media with low OHM dissipation) for the

boundary conditions E = 0 and H = 0 for t £ 0:

Ñ´ =e +s

H E (1.54)

-Ñ´ =m +

E H (1.55)

eÑgE = 0 (1.56)

mÑgH = 0 (1.57)

with

s: specific magnetic conductivity [V / Am]

In the interpretation of this ansatz MEYL has gone again a step further and declares the

equations (1.54) to (1.57) to be valid in all cases, what says, that there exist no kind of monopoles,

whether electric nor magnetic. The alleged electric monopoles (charges) are then only

secondary effects of electric and magnetic fields.

From (1.54) to (1.57) HARMUTH[5]-GL.21 has derived the electric field equation to

( )

2 s s 0

t t

¶ ¶

D -me - ms+e - s =

¶ ¶

E E

E E (1.58)

and has shown, that this equation can be solved for a certain set of boundary conditions. The

same equation (1.58) is designated by Meyl as the fundamental field equation.

The MÚNERA-GUZMÁN-Ansatz

Héctor MÚNERA and Octavio GUZMÁN[19] have proposed the following equations (w º ct):

¶ p

Ñ´ = +

¶w

N J (1.59)

¶ p

Ñ´ =- +

¶w

P J (1.60)

ÑgN = 4pr (1.61)

ÑgP = -4pr (1.62)

with

N º B-E (1.63)

P º B + E (1.64)

From this follows MAXWELL‘s equations (1.26)-(1.29) as shown below:

FARADAY‘s law (1.26): (1.60) - (1.59) (1.65)

AMPÈRE‘s law (1.27): (1.60) + (1.59) (1.66)

COULOMB‘s law (1.28): (1.62) - (1.61) (1.67)

(1.62) + (1.61) (1.68)

In this notation the current density J and the charge density r are not understood as electric

only but merely as electromagnetic entities. With an analysis of MÚNERA and GUZMÁN it can

be shown, that beneath the electric scalar field also a non-trivial magnetic scalar field should

exist.

Imaginary Expansions of Maxwell‘s Equations

The Notation in Minkowski-Space

In electrodynamics the relativistic notation is fully established. Because of the second

EINSTEIN‘ian postulate[5] about the absolute constancy of the speed of light (therefore its

independency of the speed of the light source or light detector) the four-dimensional notation

has been developed. But the force field vectors E and H can not be used for four-vectors. But

the potentials and the charge densities have been regarded as very optimal to formulate the

electrodynamics in a compact form. If we first have an event vector

X = ict + x

then it follows in MINKOWSKI-Space the invariance of the four-dimensional length ds

2 2

1 1 2 2 3 3 ds dx dx dx dx dx dx dx dx c dt m m = = + + -

and

2 ( )

2 1 1 2 2 3 3

i 1

d dx dx dt dx dx dx dx dx dx

c c m m t = = - + +

From this follows the four-dimensional velocity vector to

( )

d 1

d u

= = +

which gives the four-dimensional current density

0 ( )

0 2

=r = +

J U u

With the four-dimensional gradient operator

1 2 3 ic t x x x

¶ ¶ ¶ ¶

Ñ= + + +

¶ ¶ ¶ ¶

i j k

follows with

4ÑgJ = 0

the continuity equation (1.8). With the four-dimensional vector potential

A = ij+ A

and with the D‘ALEMBERT operator

( ) 2

4 2 2 2

2 2 2 2

1 1

c t x x c t n n

¶ ¶ ¶

Ñ = =Ñ- = -

¶ ¶ ¶ ¶ W

follows the relation

2 1

W A = J

But then the possibility for a compact and easy calculation within the MINKOWSKI-Space

comes to an end. To include the electric and magnetic fields, the following definition

A A

x x

n m

m n

¶ ¶

º -

¶ ¶

is used to determine the electromagnetic field tensor:

3 2 1

3 1 2

2 1 3

1 2 3

0 B B iE

B 0 B iE

B B 0 iE

iE iE iE 0

ì - - ü

ï- - ï ï ï =í

- - ý ï ï

îï ïþ

With two equations with the components of the field tensor the four MAXWELL equations can

be derived. With the first equation

F F F

x x x

lm mn nl

n l m

¶ ¶ ¶

+ + =

¶ ¶ ¶

follows for an arbitrary combination of l, m, n to 1, 2, 3 the MAXWELL equation (1.29)

12 23 31 3 1 2

3 1 2 3 1 2

F F F B B B

0 0

x x x x x x

¶ ¶ ¶ ¶ ¶ ¶

+ + = ® + + =Ñ =

¶ ¶ ¶ ¶ ¶ ¶

and if one of the indices l, m, n is equal 4 it follows the MAXWELL equation (1.27). With the

second equation

F 1

x c

follow the non-homogenous MAXWELL equations (1.26) and (1.28).

Simple Complex Notation

One possibility to enhance the symmetry of MAXWELL‘s equations offers the inclusion of

imaginary numbers. INOMATA[13] uses the imaginary axis only for the „missing“ terms in

MAXWELL‘s equations. Thus they become:

m m

i i

= e Ñ = r Ñ´ = +

= m Ñ =r -Ñ´ = +

D E D H j

B H B E j

(1.69)

From this result an imaginary magnetic charge and an imaginary magnetic current density. In

this notation the imaginary unit i is used for variables, which are not physically existent (i.e.

are not measurable until now). Thus by using „i“ in the equations above the missing variables

are placed into an imaginary (non existent) space i(x1, x2, x3).

Eight-dimensional, Complex Notation

Elizabeth RAUSCHER[19] proposes a consequent expansion of the complex notation, so that for

each field and for each charge density a real and an imaginary part is introduced.

Re Im

Re Im e m

= +

= + = +

r =r +r =r +r

E E E

B B B

j j j j j

(1.70)

Then, when using a correct splitting of the terms, two complementary sets of MAXWELL

equations can be formulated. The real equations are:

Re Re Re Re Re Re

Re Re Re Re

= e Ñ = r Ñ´ = +

= m Ñ = -Ñ´ =

D E D H j

B H B E

(1.71)

With an elimination of i on both sides we get for the imaginary parts:

Im t

Im Im Im Im

Im Im Im Im Im

= e Ñ = Ñ´ =

= m Ñ =r -Ñ´ = +

D E D H

B H B E j

(1.72)

As used by INOMATA also RAUSCHER uses the imaginary unit „i“ to sort the physical existent

variables from the physical non existent ones.

The Imaginary Quaternion Notation

An other possibility is the mixture of quaternions and imaginary numbers, what has for

example be done by HONIG[12]. With the vector potential and the current density

4 x y z

i A A A

i v v v

i j k

= j+ + +

= r+r +r +r

(1.73)

follows with the operator

q i

t x x x

i j k

¶ ¶ ¶ ¶

= + + +

¶ ¶ ¶ ¶ W (1.74)

and with the LORENTZ condition q 4 W gA = 0 the MAXWELL equations with

4 3 4 i i i i

t t

æ ¶ ¶ ö = Ñ +Ñ +Ñ´ +Ñ´ + ç- + ÷ = r + = è ¶ ¶ ø

E B

W A g E gB E B J J (1.75)

Actually this notation is very efficient. It is, for example, easily possible to formulate the

LORENTZ force or equations of the quantum electrodynamics with this notation. But now the

imaginary unit „i“ is not used to separate the observable variables from the non existent ones.

Interestingly there does not exists one single real number at all. Each real number is associated

either with the imaginary unit „i“ or with the HAMILTON units i, j and k. With some

additional rules also this notation can be expanded to eight dimensions. This should be

presented in another paper.

Closing Remarks

Different notations to the MAXWELL equations are presented. Depending on the application

one or another notation can be very useful, but at the end the presented variety is not satisfactory.

This variety can be a hint, that the correct final form has not been found until now.

Many discussions have been presented about the existence of magnetic monopoles. But

either the electric field is only a subjective measuring caused by the relative motion between

charges -–as it is said by the Special Theory of Relativity – or the magnetic force field can be

derived from a scalar potential field. In the first case magnetic monopoles can not exist, in the

second case they can exist. Despite of extensive experiments no magnetic monopoles have

been found until now. So we can conclude, that no magnetic potential fields must be postulated

and that the non symmetry in MAXWELL‘s equations still are correct.

Proposals to enhance the symmetry with imaginary numbers are interesting but covers the

danger, that with the simple mathematical tool „i“ a symmetric formulation can be reached

vastly, but that the physical models do become nebulous.

References

[1] AHARONOV Yakir & David BOHM, „Significance of Electromagnetic Potentials in the Quantum

Theory”, Physical Review 115 /3 (01 August 1959)

[2] BARRETT Terence W., “Comments on the HARMUTH ansatz: Use of a magnetic current density in

the calculation of the propagation velocity of signals by amended Maxwell theory”, IEEE Trans.

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Physics 34 (1966) 202-211

[4] DIRAC Paul André Morice, „Quantised Singularities in the Electromagnetic Field“, Proceedings

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421

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3153-3159

[15] MAXWELL James Clerk, „A Dynamical Theory of the Electromagnetic Field”, Royal Society

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[16] MAXWELL James Clerk, „A Treatise on Electricity & Magnetism“, (1873) Dover Publications,

New York ISBN 0-486-60636-8 (Vol. 1) & 0-486-60637-6 (Vol. 2)

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