Electromagnetic Classical Field Theory in a Form Independent of Specific Units
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Electromagnetic Classical Field Theory in a Form Independent of Specific Units
Francesco F. Summa ∗ School of Engineering, University of Basilicata, 85100 Potenza Italy
In this article we have illustrated how is possible to formulate Maxwell’s equations in vacuum inan independent form of the usual systems of units. Maxwell’s equations, are then specialized to themost commonly used systems of units: International system of units (SI), Gaussian normal, Gaus-sian rational (Heaviside-Lorentz), C.G.S. (electric), C.G.S. (magnetic), natural normal and naturalrational. Both, the differential and the integral formulations of Maxwell’s equations in vacuum, areillustrated. Also the covariant formulation of Maxwell’s equation is illustrated.
I. INTRODUCTION
Usually, in literature and in many texts, Maxwell’s equations are expressed in different systems of units. This very oftenleads to a great confusion and to a mixed intermediate treatment between the various systems of units. The key idea,developed in this article, is to specialize Maxwell’s equations for a generic system of units, showing step by step how ispossible to derive the fundamental relations of classical electrodynamics. A common strategy, to accomplish this idea,consists in the introduction of a number of unspecified constants into Maxwell’s equations. For example Gelman, in hisarticle [1] , introduced five constants into Maxwell’s equations which specialize to obtain these equations in Gaussian,international system (SI), Heaviside–Lorentz (HL), C.G.S. (electric) and C.G.S. (magnetic) units. Similarly, Jacksonused in the second edition of his textbook [2] the Gaussian units, while introduced four constants into Maxwell’sequations which properly specialize to yield these equations in the above-mentioned units in the third edition [3] . Itis possible to demonstrate that three constants k , k and k are sufficient to express Maxwell’s equations in a wayindependent of units. In the following table we display the values of k , k and k corresponding to Internationalsystem of units (SI), Gaussian normal, Gaussian rational (Heaviside-Lorentz), natural normal, natural rational, C.G.S.(electric) and C.G.S. (magnetic). System of Units k k k SI k k m c c Heaviside-Lorentz π πc c C.G.S. electric 1 c c h = c =
1) normal 1 1 1rational π π TABLE I: The k , k and k system. The constant k is defined as k = πε while k m stands for k m = µ π Using the previous table is possible to classify the different systems of units adopted in many textbooks. For example,in the context of the SI system of units we found the Vanderline’s [4] , Bo Thidé’s [5] , Panofsky’s [6] and Griffiths [7] textbooks, while the Landau’s [8] textbook adopts Gaussian normal units, Cohen’s [9] textbook adopts natural normalunits and Barut’s [10] textbook adopts natural rational units. ∗ Electronic address: [email protected]
II. DIFFERENTIAL FORMULATION OF MAXWELL’S EQUATIONS
The electromagnetic field in vacuum is described by Maxwell’s equations, that govern the dynamics of the electricfield E and the magnetic field B : ∇ · E = πk ρ (2.1) ∇ × B = πk J + ( k k ) ∂ E ∂t (2.2) ∇ · B = ∇ × E = − k ∂ B ∂t (2.4)The first and the third equations are homogeneous and correspond respectively to the Gauss law for the electric fieldand the magnetic one. The last two equations are the Ampere-Maxwell and the Faraday-Neumann-Lenz laws. Theyare not homogeneous and hold the charge density ρ and the density current vector J . From Maxwell’s equation weextract the continuity equation. To do this, we have to consider only the inhomogeneous equations: ∇ · E = πk ρ ∇ × B = πk J + ( k k ) ∂ E ∂t (2.5)If we apply the operators ∂∂t to the first equation and the ∇· to the second one we obtain ∂∂t ( ∇ · E ) = πk ∂ρ∂t ∇ · h ∇ × B − ( k k ) ∂ E ∂t i = πk ( ∇ · J ) (2.6)Now we know that for a generic vector a we have ∇ · ( ∇ × a ) = ∂∂t ( ∇ · E ) = πk ∂ρ∂t −( k k ) ∇ · ( ∂ E ∂t ) = πk ( ∇ · J ) (2.7)Since the derivation order is indifferent ∂∂t ∂∂x α = ∂∂x α ∂∂t we have ∇ · ( ∂ E ∂t ) = πk ∂ρ∂t − ∇ · ( ∂ E ∂t ) = πk ( ∇ · J ) (2.8)If we add the two equations to each other, we obtain the continuity equation ∂ρ∂t + ∇ · J = ρ and J and expresses the law of conservation of the electric charge. The continuity equation is unitindependent. The meaning of this equation became clear if we rewrite it in an integral form. Integrating both memberson a volume V we have: ZZZ V ∇ · J dV = − dd t ZZZ V ρ dV (2.10)where, taking into account that the only quantity dependent on t is the charge density ρ , we have taken the derivativewith respect to time out of the integral sign and wrote it as a total derivative. Using now the Gauss’s theorem totransform the volume integral to the first member into an integral on the closed surface Σ which delimits V we obtain ✞✝ ☎✆ ZZ Σ J · n dΣ = − dd t ZZZ V ρ dV (2.11)This equation can be rewritten as I = − dQdt (2.12)where Q is the total charge contained in V and I is the current that flows through Σ . If Q increases there is a negativecurrent flow, that is a certain amount of charge enters V and vice versa. In general, in an electrodynamic problem,the Lorentz’s force is introduced to account for charge particles. This force can be obtained by defining a Lorentz’sdensity ρ L = ρ E + k ( J × B ) (2.13)The force exerted by the field on the entire charge distribution is given by the integral on the whole volume: F L = ZZZ V ρ L dV = ZZZ V ρ E + k ( J × B ) dV (2.14)However, there is an important effect to be taken into consideration that we will not consider. Indeed a charged,accelerating particle, emits electromagnetic radiation which feeds back on it, affecting its motion. This effect is calleda radiation reaction and can be considered negligible if the speed variation over time, therefore the acceleration,is sufficiently small. Now we want to derive the field equations for B and E . To do this we derive from time theFaraday-Neumann-Lenz equation ∂∂t ( ∇ × E ) = − k ∂ B ∂t (2.15)and because the derivative order is not important we can write ∇ × ∂ E ∂t = − k ∂ B ∂t (2.16)Now using the Ampere-Maxwell law we can write the following expression ∂ E ∂t = k k ( ∇ × B − πk J ) (2.17)that have to be replaced in the previous equation to obtain ∇ × (cid:20) ( k k )( ∇ × B ) − πk J (cid:21) = − k ∂ B ∂t (2.18)that can be rewritten as ( k k ) ∇ × ( ∇ × B ) − πk ( ∇ × J ) = − k ∂ B ∂t (2.19)After some steps and using the relation ∇ × ( ∇ × a ) = ∇ ( ∇ · a ) − ∇ a for a generic vector a we can write ( k k ) (cid:2) ∇ ( ∇ · B ) − ∇ B (cid:3) − πk ( ∇ × J ) = − k ∂ B ∂t (2.20)being ∇ · B = − ∇ B − πk ( ∇ × J ) + ( k k k ) ∂ B ∂t = (cid:3) we have to define k k k = c , in this waywe obtain (cid:3) B = − πk ( ∇ × J ) (2.22)A similar procedure can be used to obtain the equation for the electric field E ∇ × ( ∇ × E ) = − k ∇ × ( ∂ B ∂t ) (2.23) ∇ ( ∇ · E ) − ∇ E = − k ∇ × ( ∂ B ∂t ) (2.24)Now using the Gauss equation for the electric field E we obtain ∇ ( πk ρ ) − ∇ E = − k ∂∂t ( ∇ × B ) (2.25)that we can write using the Ampere-Maxwell law as4 πk ∇ ρ − ∇ E = − k ∂∂t ( πk J + k k ∂ E ∂t ) (2.26)After some algebraic steps we obtain 4 πk ∇ ρ − ∇ E = − πk k ∂ J ∂t − k k k ∂ E ∂t (2.27)that can be rewritten as − k k k ∂ E ∂t + ∇ E = πk ∇ ρ + πk k ∂ J ∂t (2.28)Using the same condition as before k k k = c we can introduce the D’Alembert operator (cid:3) to obtain the followingequation for the electric field E (cid:3) E = π ( k ∇ ρ + k k ∂ J ∂t ) (2.29)A more manageable formulation of Maxwell’s equations is obtained by the introduction of a scalar potential ϕ anda vector potential A . We know that the magnetic field B has no divergence, so there exists a function A ( r , t ) , calledvector potential, such that: B = ∇ × A (2.30)If we replace this relationship in the Faraday-Neumann-Lenz law, we will note that there must exist a function calledscalar potential or electrical potential such that: ∇ × E + k ∂ ( ∇ × A ) ∂t = ∇ × ( E + k ∂ A ∂t ) = E + k ∂ A ∂t is irrotational [11] so we will have: E = − ∇ ϕ − k ∂ A ∂t (2.33)Using the expression 2.33 in the Gauss equation for the electric field we obtain: ∇ ϕ + k ∇ · ∂ A ∂t = − πk ρ (2.34)If we substitute the equations 2.30 and 2.33 in the Ampere-Maxwell law we obtain ∇ × ( ∇ × A ) = πk J + ( k k ) ∂∂t (cid:20) − ∇ ϕ − k ∂ A ∂t (cid:21) (2.35)Using the previous relation for a generic vector a ∇ × ( ∇ × a ) = ∇ ( ∇ · a ) − ∇ a we can write ∇ ( ∇ · A ) − ∇ A = πk J − ( k k k ) ∂ A ∂t − ( k k ) ∇ ∂ϕ∂t (2.36)that after some algebraic passages can be written as (cid:3) A − ∇ (cid:20) ∇ · A + ( k k ) ∂ϕ∂t (cid:21) + πk J = ∇ · A + ( k k ) ∂ϕ∂t = ∇ · A = ϕ =
0, the Coulomb gauge instead establishes that ∇ · A = ϕ = III. INTEGRAL FORMULATION OF MAXWELL EQUATIONS
Now we can see how is possible to obtain the integral version of Maxwell’s equations. We pick any region V we wantand integrate both sides of each equation over that region: ZZZ V ∇ · E dV = ZZZ V πk ρ dV (3.1) ZZZ V ∇ · B dV = ✞✝ ☎✆ ZZ Σ E · n dΣ = πk Q (3.3) ✞✝ ☎✆ ZZ Σ B · n dΣ = Q = P ni = q i is the total charge contained within the region V and Σ = ∂V . Gauss law tells us that the fluxof the electric field out through a closed surface is (basically) equal to the charge contained inside the surface, whileGauss law for magnetism tells us that there is no such thing as a magnetic charge. For Faraday’s law we pick anysurface Σ and integrate the flux of both sides through it: ✞✝ ☎✆ ZZ Σ ( ∇ × E ) · n dΣ = ✞✝ ☎✆ ZZ Σ − k ∂ B ∂t · n dΣ (3.5)On the left we can use Stokes theorem, while on the right we can pull the derivative outside the integral: I ∂Σ E · ds = − k ∂∂t Φ Σ ( B ) (3.6)where Φ Σ ( B ) is the flux of the magnetic field B through the surface Σ . Faraday’s law tells us that a changing magneticfield induces a current around a circuit. A similar analysis helps with Ampere’s law: ∇ × B = πk J + ( k k ) ∂ E ∂t (3.7)We pick a surface and integrate: ✞✝ ☎✆ ZZ Σ ( ∇ × B ) · n dΣ = ✞✝ ☎✆ ZZ Σ πk J · n dΣ + ✞✝ ☎✆ ZZ Σ ( k k ) ∂ E ∂t · n dΣ (3.8)Then we simplify each side: I ∂Σ B · ds = πk I Σ + ( k k ) ∂∂t Φ Σ ( E ) (3.9)where Φ Σ ( E ) is the flux of the electric field E through the surface Σ , and I Σ is the total current flowing through thesurface Σ . Ampere’s law tells us that a flowing current induces a magnetic field around the current, and Maxwell’scorrection tells us that a changing electric field behaves just like a current made of moving charges. We collect thesetogether into the integral form of Maxwell’s equations: ✞✝ ☎✆ ZZ Σ E · n dΣ = πk Q (3.10) ✞✝ ☎✆ ZZ Σ B · n dΣ = I C E · ds = − k ∂∂t Φ Σ ( B ) (3.12) I C B · ds = πk I Σ + ( k k ) ∂∂t Φ Σ ( E ) (3.13)where C = ∂Σ . IV. ENERGY CONSERVATION
We consider a system of fields and particles contained in a volume V . We can state that, if the sum of the energyassociated with the electromagnetic fields in V , increases then there is a flow of electromagnetic energy from theoutside to the inside and vice versa. In mathematical terms this law translates into the following equation:d U em d t = − Φ em (4.1)where U em is the energy of the electromagnetic field and Φ em is the flow of the electromagnetic energy through thesurface Σ that contains the volume V . Introducing the energy density of the electromagnetic field u em and the flowof electromagnetic energy per unit of surface P , we will have: U em = ZZZ V u em dV (4.2) Φ em = ✞✝ ☎✆ ZZ Σ P · n dΣ (4.3)we obtain d U em d t = − dd t ZZZ V u em dV − ✞✝ ☎✆ ZZ Σ P · n dΣ (4.4)Now using the Gauss theorem we obtaindd t ZZZ V u em dV = − ZZZ V ∇ · P dV (4.5)Being fixed the domain of integration we can bring the derivative in the sign of integral replacing it with a partial one ZZZ V (cid:20) ∂u em ∂t + ∇ · P (cid:21) dV = ∂u em ∂t + ∇ · P = u em and the Poynting vector P as a function of the fields,starting from the Ampere-Maxwell and Farday-Neumann-Lenz equations (cid:14) ∇ × B = πk J + ( k k ) ∂ E ∂t ∇ × E = − k ∂ B ∂t (4.8)Multiply by scaling the first equation for E and the second for B we obtain (cid:14) E · ( ∇ × B ) = πk E · J + ( k k ) E · ∂ E ∂t B · ( ∇ × E ) = − k B · ∂ B ∂t (4.9)Subtracting member to member we get E · ( ∇ × B ) − B · ( ∇ × E ) = πk E · J + ( k k ) E · ∂ E ∂t + k B · ∂ B ∂t (4.10)then using vector notation ∇ · ( a × b ) = b · ( ∇ × a ) − a · ( ∇ × b ) we obtain − ∇ · ( E × B ) = πk E · J + ( k k ) E · ∂ E ∂t + k B · ∂ B ∂t (4.11)that can be rewritten as 14 π (cid:20) E k · ∂ E ∂t + ( k k ) B · ∂ B ∂t (cid:21) + ∇ · ( E × B ) πk = − J · E (4.12)Now if we note that ∂ E ∂t = E · ∂ E ∂t (4.13) ∂ B ∂t = B · ∂ B ∂t (4.14) ∂∂t ( E + B ) = ( E · ∂ E ∂t + B · ∂ B ∂t ) (4.15)we can rewrite the previous equation as18 π ∂∂t (cid:20) E k + ( k k ) B (cid:21) + ∇ · ( E × B ) πk = − J · E (4.16)which compared to the equation 4.7 allows us to define the energy density of the electromagnetic field and the Poyntingvector as u em = π (cid:20) E k + ( k k ) B (cid:21) (4.17) P = E × B πk (4.18)In our case this is possible because the particles are absent so the current density J is zero. V. MOMENTUM CONSERVATION
We consider the Gauss law for the electric field and the Ampere-Maxwell law: (cid:14) ∇ · E = πk ρ ∇ × B = πk J + ( k k ) ∂ E ∂t (5.1)We multiply the first equation by E and the second by B × , we obtain (cid:14) E ( ∇ · E ) = πk ρ EB × ( ∇ × B ) = πk ( B × J ) + ( k k ) B × ∂ E ∂t (5.2)Using the vector product’s anticommutative property we can rewrite the second equation as B × ( ∇ × B ) = − πk ( J × B ) + ( k k ) B × ∂ E ∂t (5.3)Now subtracting member to member the two equations we get E ( ∇ · E ) − B × ( ∇ × B ) = πk ρ E + πk ( J × B ) − ( k k ) B × ∂ E ∂t (5.4)which after some steps can be rewritten as14 π (cid:12) E ( ∇ · E ) − B × ( ∇ × B ) + ( k k ) B × ∂ E ∂t (cid:13) = k ρ E + k ( J × B ) (5.5)Now considering that ∂∂t ( E × B ) = E × ∂ B ∂t + ∂ E ∂t × B (5.6)we obtain ∂ E ∂t × B = ∂∂t ( E × B ) − E × ∂ B ∂t (5.7)that can be rewritten using the anticommutative property of vector product as − B × ∂ E ∂t = ∂∂t ( E × B ) − E × ∂ B ∂t (5.8)Using the previous relation and considering from Faraday’s law that ∂ B ∂t = − k − ( ∇ × E ) (5.9)we can rewrite equation 5.5 as π (cid:12) E ( ∇ · E ) − B × ( ∇ × B ) + ( k k ) (cid:20) − k − E × ( ∇ × E ) − ∂∂t ( E × B ) (cid:21) (cid:13) = k ρ E + k ( J × B ) (5.10) This relation can be integrated. Now we can define the rate of change of the particle’s momentum in an electromagneticfield as d P m d t = π ZZZ V [ k ρ E + k ( J × B )] dV (5.11)so we obtain d P m d t = π ZZZ V (cid:20) E ( ∇ · E ) − B × ( ∇ × B ) − ( k k k ) E × ( ∇ × E ) (cid:21) dV − k πk ∂∂t ZZZ V ( E × B ) dV (5.12) where we may identify the second integral on the right as the electromagnetic momentum in the volume V : k πk ZZZ V ( E × B ) dV (5.13)The integrand can be interpreted as a density of electromagnetic momentum Π em = k πk ( E × B ) (5.14)We note that this momentum density is proportional to the Poynting vector P , with proportionality constant k k . VI. TOTAL ANGULAR MOMENTUM AND ITS DECOMPOSITION
Having defined the moment density, it is now possible to define an angular momentum density as L em = r × Π em (6.1)which explicitly takes the following form L em = r × k πk ( E × B ) (6.2)Now using the relation B = ∇ × A we can rewrite L em as L em = r × k πk [ E × ( ∇ × A )] (6.3)From the previous equation is possible to define the total angular momentum J em as J em = ZZZ V r × k πk [ E × ( ∇ × A )] dV (6.4)Now if we consider the vector identity E × ( ∇ × A ) = X m E m ∇ A m − ( E · ∇ ) A (6.5)we can write the total angular momentum as J em = k πk X m ZZZ V E m ( r × ∇ ) A m dV − k πk ZZZ V r × ( E · ∇ ) A dV (6.6)where the first term corresponds to orbital angular momentum and the second term is to be manipulated into theform of spin angular momentum, which does not depend linearly on r . Of course, the decomposition is meaningless ifthe gauge of A is not fixed. The gauge that is invariably chosen in this situation is the Coulomb gauge. The secondterm of equation 6.6 is treated in the following manner. We construct the vector VV = X m ∂∂x m ( E m r × A ) (6.7)or V | i = X mjk ε ijk ∂∂x m ( E m r j A k ) (6.8)where ε ijk is the Levi-Civita symbol. By using the identity ∂∂x m ( E m r j A k ) = δ jm E m A k + r j ( A k ∂E m ∂x m + E m ∂A k ∂x m ) (6.9)we get V | i = X jk ε ijk E j A k + X mjk ε ijk r j ( A k ∂E m ∂x m + E m ∂A k ∂x m ) (6.10)or V = E × A + r × A ( ∇ · E ) + r × ( E · ∇ ) A (6.11)Integrating over the dV and with the assumption that N = ZZZ V V dV = J em = k πk ZZZ V " X m E m ( r × ∇ ) A m + E × A + r × A ( ∇ · E ) dV = L + S + k πk ZZZ V r × A ( ∇ · E ) dV (6.13) The first term L is denoted the orbital term, the second term S is the spin term and the third term is non-zero onlyin the presence of charge.0 VII. COVARIANT FORMULATION OF MAXWELL’S EQUATIONS
In this section we adopt the Einstein’s summation convention on repeated indices. To express a covariant formulationof Maxwell’s equations is necessary to introduce the concept of four-vector and the metric signature of the Minkowskispacetime g µv . A four-vector in spacetime can be represented in the relativistic notation as F µ = ( f , F ) , where f isits time component and F is space component. The metric signature of the Minkowski spacetime is g µv = g µv = − − − (7.1)Derivatives in spacetime are defined by ∂ µ = ∂∂x µ = (cid:2) c ∂∂t , ∇ (cid:3) and ∂ µ = ∂∂x µ = (cid:2) c ∂∂t , − ∇ (cid:3) . The source of theelectromagnetic field tensor F µv is the four-current J µ = ( cρ , J ) (7.2)The electromagnetic field tensor F µv satisfies the Maxwell’s equations in k , k and k units: ∂ γ F µv + ∂ µ F vγ + ∂ v F γµ = ∂ µ F µv = πk J v (7.4)where the first equation stands for the homogeneous Maxwell’s equations and come from the definition of an antisym-metric tensor, while the second must derive from the Lagrangian density of the electromagnetic field and refers to thenon-homogeneous equations. We have to note that the covariant expression of the homogeneous Maxwell’s equationsis more simple if we introduce the concept of the dual of F µv , defined as F ∗ µv = ε µvkσ F kσ . Using the definition of F ∗ µv the homogeneous Maxwell’s equations can be written as ∂ µ F ∗ µv =
0. The tensor F µv is defined as F µv = − k k cE x − k k cE y − k k cE zk k cE x − B z B yk k cE y B z − B xk k cE z − B y B x (7.5)We define a vector polar if the sign of its components changes if we reverse the direction of the Cartesian axes whileaxial a vector that does not enjoy this property. Using this definitions we call F i and F ij the polar and the axialcomponents of F µv defined respectively as F i = k k c ( E ) i (7.6) F ij = − ε ijk ( B ) k (7.7)with ε ijk the Levi-Civita symbol, ( E ) i and ( B ) k that represent the components of the electric and magnetic fields.The components of the dual tensor F ∗ µv can be obtained from those of F µv by making the following changes: F ∗ i → ( B ) i (7.8) F ∗ ij → k c ε ijk ( E ) k (7.9)where we have used the relation k k c = k c . With the aid of the above definition, we can write the following four-vectorsin the ( + ) notation as: ∂ µ F µv = ( k k c ∇ · E , ∇ × B − k k ∂ E ∂t ) (7.10) ∂ µ F ∗ µv = ( ∇ · B , − ∇ × E − k ∂ B ∂t ) (7.11)The four-potential is defined in the ( + ) notation as A µ = ( k k cϕ , A ) (7.12)1In order for the equation 7.4 to be verified, the Lagrangian density must have the following form L = − k πk F µv F µv − k J µ A µ (7.13)that can be rewritten as L = k πk ( ∂ µ A v − ∂ v A µ )( ∂ µ A v − ∂ v A µ ) − k J µ A µ (7.14)or in non manifest covariant notation as L = π " (cid:12)(cid:12) E (cid:12)(cid:12) k − k k (cid:12)(cid:12) B (cid:12)(cid:12) − ρϕ + k J · A (7.15)To show that starting from the Lagrangian density, defined above, which is manifestly covariant or not, only two ofthe four Maxwell’s equations are obtained, we apply the Euler Lagrange fields equations, defined for a generic field ψ j as X j " ∂ L ∂ψ j − ∂∂t ( ∂ L ∂ ( ∂ψ j ∂t ) ) − X k = ∂∂x k ( ∂ L ∂ ( ∂ψ j ∂x k ) ) = E and B , as function of potentials, by doing so we get L = π (cid:20) ( ∇ ϕ ) k + k ∇ ϕk ∂ A ∂t + k k ( ∂ A ∂t ) − k k ( ∇ × A ) (cid:21) − ρϕ + k J · A (7.17)In this context the fields defined in the equations 7.16 are ϕ and A . Applying Euler Lagrange’s equations to ϕ we get ∂ L ∂ ( ∂ϕ∂x k ) = − E k πk (7.18) ∂ L ∂ ( ∂ϕ∂t ) = ∂ L ∂ϕ = − ρ (7.20)from which we obtain ∇ · E = πk ρ (7.21)We now carry out a similar procedure for A . For sake of simplicity we consider only the component A x , we get ∂ L ∂ ( ∂A x ∂t ) = − k πk E x (7.22) ∂ L ∂ ( ∂A x ∂x ) = k πk B z (7.23) ∂ L ∂ ( ∂A x ∂x ) = − k πk B y (7.24) ∂ L ∂ ( ∂A x ∂x ) = ∂ L ∂A x = k J x (7.26)from which we derive k πk ( ∂B z ∂x − ∂B y ∂x ) − k πk ∂E x ∂t − k J x = ∇ × B = πk J + ( k k ) ∂ E ∂t (7.28)From the previous relations it is easy to derive the Hamiltonian density of our system H = X j π j ∂ψ j ∂t − L (7.29)considering that the canonical momentum densities π j are defined as π j = ∂ L ∂ ( ∂ψ j ∂t ) (7.30)Indeed we have π A = ∂ L ∂ ( ∂ A ∂t ) = − k πk E = k πk (cid:20) ∇ ϕ + k ∂ A ∂t (cid:21) (7.31) π ϕ = ∂ L ∂ ( ∂ϕ∂t ) = H = π (cid:20) k k ( ∂ A ∂t ) + k k ( ∇ × A ) − ( ∇ ϕ ) k (cid:21) + ρϕ − k J · A (7.33)If we want to obtain all Maxwell’s equations by apply the Euler Lagrange fields equations, we have to define a newLagrangian density. This theoretical problem is mentioned in the Baker’s article [12] and it is not the aim of this paper. VIII. DISCUSSION
Graduate students actually work with equations in both SI and Gaussian units, which does not seem to be the bestalternative from a pedagogical point of view. For this reason the k k k system may be a pedagogical alternative forundergraduate and graduate students who can solve electromagnetic problems without having to work in a specificsystem of units. IX. CONCLUSIONS
In this paper we have considered the old idea of writing electromagnetic equations in a form independent of specificunits showing step by step how it is possible to derive the fundamental relations of classical electrodynamics. We havefollowed the work of Jackson [3] , that introduce four constants in Maxwell’s equations, showing that only three areneeded. We have shown that for each system of units the relationship k k k = c must occur to define the D’Alembertoperator. This paper can be used as an introductive chapter in many courses in order to clarify all the mathematicalderivations, performed step by step, that lead to the different formulations of the Classical Electromagnetic FieldTheory, also in the covariant notation adopted in Special Relativity. [1] H. Gelman, Generalized Conversion of Electromagnetic Units, Measures, and Equations, Am. J. Phys. 34 291 (1966).[2] J. D. Jackson, Classical Electrodynamics, 2nd edn. Wiley (1962).[3] J. D. Jackson, Classical Electrodynamics, 3rd edn. Wiley (1999).[4] J. Vanderline, Classical Electromagnetic Theory (New York: Wiley) (1993).[5] Bo Thidé, Electromagnetic Field Theory, 2nd ed. Dover (2011).[6] W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company,Inc., Reading, MA...,(1962).[7] D. J. Griffiths, Introduction to Electrodynamics (Pearson, 2013).3