Helmholtz’s theorem for two retarded fields and its application to Maxwell’s equations

The mathematical foundations of electrostatics and magnetostatics relies on the Helmholtz theorem of vector analysis. Among the several equivalent formulations of this theorem presented in standard textbooks [1, 2, 3, 4], let us consider the formulation given by Griffiths [1]. The theorem states that if the divergence $\nabla\cdot{\bf F}=D({\bf r})$ and the curl $\nabla\times{\bf F}={\bf C}({\bf r})$ of a vector function ${\bf F}({\bf r})$ are specified, and if they both go to zero faster than $1/r^{2}$ as $r\rightarrow\infty,$ and if ${\bf F}({\bf r})$ goes to zero as $r\rightarrow\infty,$ then ${\bf F}$ is uniquely given by ${\bf F}=-\nabla U+\nabla\times{\bf W},$ where

$\displaystyle U({\bf r})\equiv\frac{1}{4\pi}\int\frac{D({\bf r}^{\prime})}{R}\,d^{3}r^{\prime},\;\;W({\bf r})\equiv\frac{1}{4\pi}\int\frac{{\bf C}({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}.$

(1)

Here ${\bf r}$ is the field point and $r=|{\bf r}|$ is its magnitude, ${\bf r}^{\prime}$ is the source point and $R=\!|{\bf r}-{\bf r}^{\prime}|$. The integrals are over all space and $d^{3}r^{\prime}$ is the volume element. This theorem has a useful corollary: Any vector function ${\bf F}({\bf r})$ that goes to zero faster than $1/r$ as $r\rightarrow\infty$ can be expressed as

$\displaystyle{\bf F}({\bf r})=-\nabla\bigg{(}\frac{1}{4\pi}\int\frac{\nabla^{\prime}\cdot{\bf F}({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}\bigg{)}+\nabla\times\bigg{(}\frac{1}{4\pi}\int\!\frac{\nabla^{\prime}\times{\bf F}({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}\bigg{)}.$

(2)

In a first application, we write ${\bf F}={\bf E}$ and use the electrostatic equations: $\nabla\cdot{\bf E}=\rho/\epsilon_{0}$ and $\nabla\times{\bf E}=0$ to obtain the electrostatic field ${\bf E}({\bf r})$ produced by the charge density $\rho({\bf r})$:

$\displaystyle{\bf E}({\bf r})=-\nabla\bigg{(}\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}\bigg{)},$

(3)

or equivalently ${\bf E}=-\nabla\Phi,$ where $\Phi$ is the scalar potential:

$\displaystyle\Phi({\bf r})=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}.$

(4)

In a second application we make ${\bf F}={\bf B}$ and use the magnetostatic equations: $\nabla\cdot{\bf B}=0$ and $\nabla\times{\bf B}=\mu_{0}{\bf J}$ to obtain the magnetostatic field ${\bf B}({\bf r})$ produced by the current density ${\bf J}({\bf r})$:

$\displaystyle{\bf B}({\bf r})=\nabla\times\bigg{(}\frac{\mu_{0}}{4\pi}\int\frac{{\bf J}({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}\bigg{)},$

(5)

or equivalently ${\bf B}=\nabla\times{\bf A},$ where ${\bf A}$ is the vector potential:

$\displaystyle{\bf A}({\bf r})=\frac{\mu_{0}}{4\pi}\int\frac{{\bf J}({\bf r}^{\prime})}{R}\,d^{3}r^{\prime}.$

(6)

Here $\epsilon_{0}$ and $\mu_{0}$ are the permittivity and permeability of vacuum which satisfy $\epsilon_{0}\mu_{0}=1/c^{2}$, with $c$ being the speed of light in vacuum.

The Helmholtz theorem is also applicable to the time-dependent regime of Maxwell’s equations [5, 6]. The reason is simple: the derivation of (2) does not involve time and therefore it can be applied to a time-dependent vector field ${\bf F}({\bf r},t)$. We simply make the replacements ${\bf F}({\bf r})\!\to\!{\bf F}({\bf r},t)$ and ${\bf F}({\bf r}^{\prime})\!\to\!{\bf F}({\bf r}^{\prime},t)$ in (2) and obtain an instantaneous form of the theorem [6]:

$\displaystyle{\bf F}({\bf r},t)\!=-\nabla\bigg{(}\frac{1}{4\pi}\int\frac{\nabla^{\prime}\cdot{\bf F}({\bf r}^{\prime},t)}{R}\,d^{3}r^{\prime}\bigg{)}+\nabla\times\bigg{(}\frac{1}{4\pi}\int\!\frac{\nabla^{\prime}\times{\bf F}({\bf r}^{\prime},t)}{R}\,d^{3}r^{\prime}\bigg{)}.$

(7)

However, this form of the theorem is of limited practical utility in the time-dependent regime of Maxwell’s equations because the Faraday law and the Amp$\grave{\rm e}$re-Maxwell law in this regime do not specify the curls of the fields ${\bf E}$ and ${\bf B}$ as such in terms of sources, rather they specify the hybrid quantities $\nabla\times{\bf E}+\partial{\bf B}/\partial t$ and $\nabla\times{\bf B}-(1/c^{2})\partial{\bf E}/\partial t$, which couple space and time variations of both fields. In the more general case in which there are magnetic monopoles, the Faraday and Amp$\grave{\rm e}$re-Maxwell laws read $-\nabla\times{\bf E}-\partial{\bf B}/\partial t=\mu_{0}{\bf J}_{g}$ and $\nabla\times{\bf B}-(1/c^{2})\partial{\bf E}/\partial t=\mu_{0}\textbf{J}_{e}$, where $\textbf{J}_{g}$ is the magnetic current density and $\textbf{J}_{e}$ is the electric current density. As may be seen, these laws connect the hybrid quantities $-\nabla\times{\bf E}-\partial{\bf B}/\partial t$ and $\nabla\times{\bf B}-(1/c^{2})\partial{\bf E}/\partial t$ with their respective electric and magnetic currents. It is clear that the time derivatives of the fields couple their corresponding curls. Let us call the first hybrid quantity the coupled curl of ${\bf E}$ and the second quantity the coupled curl of ${\bf B}$.

It would be desirable to have an extension of the Helmholtz theorem which allows us to directly find the retarded electric and magnetic fields in terms of the retarded scalar and vector potentials, in the same form that the standard Helmholtz theorem allows us to directly find the electrostatic and magnetostatic fields in terms of their respective static scalar and vector potentials. Expectably, this appropriate generalisation of the theorem should be formulated for two retarded vector fields and in terms of their respective divergences and coupled curls.

In this paper we formulate a generalisation of the Helmholtz theorem for two retarded vector functions. We show how a corollary of this theorem allows us to find the retarded electric and magnetic fields and introduce the corresponding retarded potentials in three specific cases: for Maxwell equations with the electric charge and current densities (the standard case), when these equations additionally have polarisation and magnetisation densities and finally when they additionally have magnetic charge and current densities. The proof of the theorem relies on the uniqueness of the solutions of the vector wave equation. A novel demonstration of this uniqueness is presented in the Appendix A. Although more complicated than the standard Helmholtz theorem, the extension of this theorem for two retarded vector fields formulated here may be presented in an advanced undergraduate course of electrodynamics.

Suppose we are told that the divergences $\nabla\cdot{\bf F}_{1}$ and $\nabla\cdot{\bf F}_{2}$ of two retarded vector functions ${\bf F}_{1}={\bf F}_{1}({\bf r},t)$ and ${\bf F}_{2}={\bf F}_{2}({\bf r},t)$ are specified by the time-dependent scalar functions $D_{1}=D_{1}({\bf r},t)$ and $D_{2}=D_{2}({\bf r},t)$:

$\displaystyle\nabla\cdot{\bf F}_{1}=D_{1},\qquad\nabla\cdot{\bf F}_{2}=D_{2},$

(8)

and that the coupled curls $-\nabla\times{\bf F}_{1}-\partial{\bf F}_{2}/\partial t$ and $\nabla\times{\bf F}_{2}-(1/c^{2})\partial{\bf F}_{1}/\partial t$ are specified by the time-dependent vector functions ${\bf C}_{1}={\bf C}_{1}({\bf r},t)$ and ${\bf C}_{2}={\bf C}_{2}({\bf r},t)$:

$\displaystyle-\nabla\times{\bf F}_{1}-\frac{\partial{\bf F}_{2}}{\partial t}={\bf C}_{2},\qquad\nabla\times{\bf F}_{2}-\frac{1}{c^{2}}\frac{\partial{\bf F}_{1}}{\partial t}={\bf C}_{1},$

(9)

where $c$ is the propagation speed of the fields ${\bf F}_{1}$ and ${\bf F}_{2}$.¹¹1Here the speed $c$ does not necessarily have to represent the speed of light. However, if $c$ is identified with the speed of light in vacuum and the following identifications: ${\bf F}_{1}={\bf E},{\bf F}_{2}={\bf B},D_{1}=\rho/\epsilon_{0},D_{2}=0,{\bf C}_{1}=\mu_{0}{\bf J}$ and ${\bf C}_{2}=0$ with $\epsilon_{0}\mu_{0}=1/c^{2}$ are made, then (8) and (9) become Maxwell’s equations in SI units. Analogously, if the identification ${\bf F}_{1}=c{\bf E},{\bf F}_{2}={\bf B},D_{1}=4\pi c\rho,D_{2}=0,{\bf C}_{1}=(4\pi/c){\bf J}$ and ${\bf C}_{2}=0$ are made then (9) and (10) become Maxwell’s equations in Gaussian units. For consistence, ${\bf C}_{1},{\bf C}_{2},D_{1}$ and $D_{2}$ must satisfy the “continuity” equations:

$\displaystyle\nabla\cdot{\bf C}_{1}+\frac{1}{c^{2}}\frac{\partial D_{1}}{\partial t}=0,\qquad\nabla\cdot{\bf C}_{2}+\frac{\partial D_{2}}{\partial t}=0,$

(10)

which are implied by (8) and (9). Question: Can we, on the basis of this information, uniquely determine ${\bf F}_{1}$ and ${\bf F}_{2}$? If $D_{1},D_{2},{\bf C}_{1}$ and ${\bf C}_{2}$ go to zero sufficiently rapidly at infinity, the answer is $yes$, as we will show by explicit construction.²²2The answer is also $yes$ for the case in which the quantities $D_{1},D_{2},{\bf C}_{1}$ and ${\bf C}_{2}$ are localised in space, that is, when they are zero outside a finite region of space. This frequently occurs in practical applications because these quantities usually play the role of sources, which are physically localised in space.

We claim that the solution of (8) and (9) is

$\displaystyle{\bf F}_{1}=-\nabla U_{1}-\nabla\times{\bf W}_{1}-\frac{\partial{\bf W}_{2}}{\partial t},\;\;\;{\bf F}_{2}=-\nabla U_{2}+\nabla\times{\bf W}_{2}-\frac{1}{c^{2}}\frac{\partial{\bf W}_{1}}{\partial t},$

(11)

where

$\displaystyle U_{1}={1\over 4\pi}\int\frac{[D_{1}]}{R}\,d^{3}r^{\prime},\qquad U_{2}={1\over 4\pi}\int{[D_{2}]\over R}]\,d^{3}r^{\prime},$

(12)

and

$\displaystyle{\bf W}_{1}={1\over 4\pi}\int{[{\bf C}_{2}]\over R}\,d^{3}r^{\prime},\qquad{\bf W}_{2}={1\over 4\pi}\int{[{\bf C}_{1}]\over R}\,d^{3}r^{\prime}.$

(13)

The integrals are over all space. Our demonstration requires two additional sets of equations. The first set is formed by the “Lorenz” conditions:

$\displaystyle\nabla\cdot{\bf W}_{1}+\frac{\partial U_{2}}{\partial t}=0,\qquad\nabla\cdot{\bf W}_{2}+\frac{1}{c^{2}}\frac{\partial U_{1}}{\partial t}=0.$

(14)

We will prove the first condition using the result [2]: $\nabla\cdot\big{(}[\mbox{\boldmath${\cal F}$\unboldmath}]/R\big{)}\!+\!\nabla^{\prime}\cdot\big{(}[\mbox{\boldmath${\cal F}$\unboldmath}]/R\big{)}\!=\![\nabla^{\prime}\cdot\mbox{\boldmath${\cal F}$\unboldmath}]/R$,

	$\displaystyle\nabla\cdot{\bf W}_{1}$	$\displaystyle=\frac{1}{4\pi}\int\nabla\cdot\bigg{(}\frac{[{\bf C}_{2}]}{R}\,d^{3}r^{\prime}\bigg{)}$			(15)
		$\displaystyle=\frac{1}{4\pi}\int\frac{[\nabla^{\prime}\cdot{\bf C}_{2}]}{R}\,d^{3}r^{\prime}-\frac{1}{4\pi}\int\nabla^{\prime}\cdot\bigg{(}\frac{[{\bf C}_{2}]}{R}\bigg{)}\,d^{3}r^{\prime}$
		$\displaystyle=-\frac{\partial}{\partial t}\bigg{(}\frac{1}{4\pi}\int\frac{[D_{2}]}{R}\,d^{3}r^{\prime}\bigg{)}-\frac{1}{4\pi}\oint\frac{\hat{{\bf n}}\cdot[{\bf C}_{2}]}{R}\,dS=-\frac{\partial U_{2}}{\partial t},$

where we have used the second equation given in (10), the property [7]: $[\partial\mbox{\boldmath${\cal F}$\unboldmath}/\partial t]=\partial[\mbox{\boldmath${\cal F}$\unboldmath}]/\partial t$, the Gauss theorem to transform the second volume integral of the second line into a surface integral ($dS$ is the surface element), which is seen to vanish at infinity by assuming the boundary condition that ${\bf C}_{2}$ goes to zero faster than $1/r^{2}$ as $r\rightarrow\infty$, and finally considering the second equation in (12).³³3In physical applications, the vector ${\bf C}_{2}$ usually represents a localised quantity. This means that ${\bf C}_{2}$ is different from zero only within a limited region of space. However, the surface of integration $S$ in the surface integral on the third line of (15) encloses all space, and therefore $S$ is outside the region where ${\bf C}_{2}$ is different from zero. It follows that ${\bf C}_{2}$ is zero everywhere on $S$ and then the surface integral vanishes. Following a similar procedure, we can prove the second “Lorenz” condition given in (14). The required second set of equations is formed by the wave equations

$\displaystyle\Box^{2}U_{1}=-D_{1},\quad\square^{2}U_{2}=-D_{2},\quad\square^{2}{\bf W}_{1}=-{\bf C}_{2},\quad\square^{2}{\bf W}_{2}=-{\bf C}_{1},$

(16)

where $\Box^{2}\equiv\nabla^{2}-(1/c^{2})\partial^{2}/\partial t^{2}$ is the d’Alembertian operator. Using the result [7]:

$\displaystyle\Box^{2}\bigg{(}\frac{[{\mbox{\boldmath${\cal X}$\unboldmath}}]}{R}\bigg{)}=-4\pi[{\mbox{\boldmath${\cal X}$\unboldmath}}]\delta({\bf r}-{\bf r}^{\prime}),$

(17)

where ${\cal X}$ represents a scalar or vector function,⁴⁴4This identity is true for functions ${\cal X}$ such that the quantities $[{\mbox{\boldmath${\cal X}$\unboldmath}}]/R$ have not the form $[{\mbox{\boldmath${\cal X}$\unboldmath}}]/R=f(R)[{\bf F}]$ with $f(R)$ being a polynomial function. If for example $f(R)=R$ then $[{\mbox{\boldmath${\cal X}$\unboldmath}}]/R=R[{\bf F}]$. It follows that $\Box^{2}(R[{\bf F}])=-4\pi R[{\bf F}]\delta({\bf r}-{\bf r}^{\prime})=0$ since the factor $R\delta({\bf r}-{\bf r}^{\prime})$ vanishes for ${\bf r}\not={\bf r}^{\prime}$ because of the delta function and also for ${\bf r}={\bf r}^{\prime}$ because this equality implies $R=0$. we can show the first wave equation in (16),

$\displaystyle\square^{2}U_{1}=\frac{1}{4\pi}\int\!\square^{2}\bigg{(}\frac{[D_{1}]}{R}\bigg{)}\,d^{3}r^{\prime}=-\int\![D_{1}]\delta({\bf r}-{\bf r}^{\prime})\,d^{3}r^{\prime}=-D_{1}.$

(18)

By an entirely similar procedure, we can show the remaining wave equations given in (16). Considering (11), (14) and (16), we can now prove that ${\bf F}_{1}$ and ${\bf F}_{2}$ satisfy (8) and (9). We take the divergences to ${\bf F}_{1}$ and ${\bf F}_{2}$ and obtain

	$\displaystyle\nabla\cdot{\bf F}_{1}$	$\displaystyle=-\nabla^{2}U_{1}-\frac{\partial}{\partial t}\nabla\cdot{\bf W}_{2}=-\nabla^{2}U_{1}+\frac{1}{c^{2}}\frac{\partial^{2}U_{1}}{\partial t^{2}}=-\Box^{2}U_{1}=D_{1},$			(19)
	$\displaystyle\nabla\cdot{\bf F}_{2}$	$\displaystyle=-\nabla^{2}U_{2}-\frac{1}{c^{2}}\frac{\partial}{\partial t}\nabla\cdot{\bf W}_{1}=-\nabla^{2}U_{1}+\frac{1}{c^{2}}\frac{\partial^{2}U_{2}}{\partial t^{2}}=-\Box^{2}U_{2}=D_{2}.$			(20)

We now calculate the coupled curls of ${\bf F}_{1}$ and ${\bf F}_{2}$ and obtain

	$\displaystyle-\nabla\times{\bf F}_{1}-\frac{\partial{\bf F}_{2}}{\partial t}=\;\nabla(\nabla\cdot{\bf W}_{1})-\nabla^{2}{\bf W}_{1}+\frac{\partial}{\partial t}\nabla\times{\bf W}_{2}$
	$\displaystyle\qquad\qquad\qquad+\nabla\bigg{(}\frac{\partial U_{2}}{\partial t}\bigg{)}-\nabla\times\frac{\partial{\bf W}_{2}}{\partial t}+\frac{1}{c^{2}}\frac{\partial^{2}{\bf W}_{1}}{\partial t^{2}}$
	$\displaystyle\qquad\qquad\quad\;\;=\;\nabla\bigg{(}\nabla\cdot{\bf W}_{1}+\frac{\partial U_{2}}{\partial t}\bigg{)}-\square^{2}{\bf W}_{1}=-\square^{2}{\bf W}_{1}={\bf C}_{2},$		(21)
	$\displaystyle\;\;\nabla\times{\bf F}_{2}-\frac{\partial{\bf F}_{1}}{\partial t}=\;\nabla(\nabla\cdot{\bf W}_{2})-\nabla^{2}{\bf W}_{2}+\frac{\partial}{\partial t}\nabla\times{\bf W}_{1}$
	$\displaystyle\qquad\qquad\qquad+\nabla\bigg{(}\frac{\partial U_{1}}{\partial t}\bigg{)}-\nabla\times\frac{\partial{\bf W}_{1}}{\partial t}+\frac{1}{c^{2}}\frac{\partial^{2}{\bf W}_{2}}{\partial t^{2}}$
	$\displaystyle\qquad\qquad\quad\;\;=\;\nabla\bigg{(}\nabla\cdot{\bf W}_{2}+\frac{\partial U_{1}}{\partial t}\bigg{)}-\square^{2}{\bf W}_{2}=-\square^{2}{\bf W}_{2}={\bf C}_{1}.$		(22)

Therefore the functions ${\bf F}_{1}$ and ${\bf F}_{2}$ given in (11) constitute a solution of (8) and (9).

Now, assuming that the conditions on $D_{1},D_{2},{\bf C}_{1}$ and ${\bf C}_{2}$ are met, is the solution in (11) $unique$? The answer is clearly $no$, for we can add to ${\bf F}_{1}$ the function ${\bf H}_{1}$ and ${\bf F}_{2}$ the function ${\bf H}_{2}$, with the divergences and coupled curls of ${\bf H}_{1}$ and ${\bf H}_{2}$ being zero, and the result still has divergences $D_{1}$ and $D_{2}$ and coupled curls ${\bf C}_{1}$ and ${\bf C}_{2}$. However, it so happens that if ${\bf F}_{1},{\bf F}_{2},\partial{\bf F}_{1}/\partial t$ and $\partial{\bf F}_{2}/\partial t$ vanish at $t=0$, and if $\nabla{\bf F}_{1},\nabla{\bf F}_{2},\partial{\bf F}_{1}/\partial t$ and $\partial{\bf F}_{2}/\partial t$ go to zero faster than $1/r^{2}$ as $r\rightarrow\infty$, and if ${\bf F}_{1}$ and ${\bf F}_{2}$ themselves go to zero as $r\rightarrow\infty$, then ${\bf H}_{1}$ and ${\bf H}_{1}$ are zero and therefore the solution (11) is unique. In the notation we are using a generic second-order tensor $\nabla\mbox{\boldmath${\cal F}$\unboldmath}$ denotes the gradient of the vector ${\cal F}$. The explicit proof that ${\bf H}_{1}$ and ${\bf H}_{1}$ are zero under their specified initial and boundary conditions is presented in Appendix A. Notice that the conditions that $\nabla{\bf F}_{1},\nabla{\bf F}_{2},\partial{\bf F}_{1}/\partial t$ and $\partial{\bf F}_{2}/\partial t$ go to zero faster than $1/r^{2}$ as $r\rightarrow\infty$ automatically imply that $D_{1},D_{2},{\bf C}_{1}$ and ${\bf C}_{2}$ go to zero faster than $1/r^{2}$ as $r\rightarrow\infty$. Now we can state the Helmholtz theorem for two retarded fields more rigorously: Theorem. If the divergences $D_{1}$ and $D_{2}$ and the coupled curls ${\bf C}_{1}$ and ${\bf C}_{2}$ of two retarded vector functions ${\bf F}_{1}$ and ${\bf F}_{2}$ are specified, and if ${\bf F}_{1},{\bf F}_{2},\partial{\bf F}_{1}/\partial t$ and $\partial{\bf F}_{2}/\partial t$ vanish at $t=0$, and if $\nabla{\bf F}_{1},\nabla{\bf F}_{2},\partial{\bf F}_{1}/\partial t$ and $\partial{\bf F}_{2}/\partial t$ go to zero faster than $1/r^{2}$ as $r\rightarrow\infty$, and if ${\bf F}_{1}$ and ${\bf F}_{2}$ themselves go to zero as $r\rightarrow\infty$, then ${\bf F}_{1}$ and ${\bf F}_{2}$ are uniquely given by (11). This theorem has the following corollary: The fields ${\bf F}_{1}$ and ${\bf F}_{2}$ can be expressed as

	$\displaystyle{\bf F}_{1}=$	$\displaystyle-\nabla\bigg{\{}\int\!\frac{[\nabla^{\prime}\cdot{\bf F}_{1}]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}}-\nabla\times\bigg{\{}\int\!\frac{[-\nabla^{\prime}\!\times{\bf F}_{1}-\partial{\bf F}_{2}/\partial t]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}}$			(23)
		$\displaystyle-\frac{\partial}{\partial t}\bigg{\{}\int\!\frac{[\nabla^{\prime}\!\times{\bf F}_{2}-(1/c^{2})\partial{\bf F}_{1}/\partial t]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}},$
	$\displaystyle{\bf F}_{2}=$	$\displaystyle-\nabla\bigg{\{}\int\!\frac{[\nabla^{\prime}\cdot{\bf F}_{2}]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}}+\nabla\times\!\bigg{\{}\int\!\frac{[\nabla^{\prime}\!\times{\bf F}_{2}-(1/c^{2})\partial{\bf F}_{1}/\partial t]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}}$			(24)
		$\displaystyle-\frac{\partial}{\partial t}\bigg{\{}\int\!\frac{[-\nabla^{\prime}\!\times{\bf F}_{1}-\partial{\bf F}_{2}/\partial t]}{4\pi R}\,d^{3}r^{\prime}\bigg{\}}.$

These expressions of the Helmholtz theorem for two retarded vector functions are considerably more complicated than the expression of the Helmholtz theorem (2) for a static vector function. However, the practical advantage (23) and (24) is that they allows us to directly find the retarded fields of Maxwell’s equations, as we will see in the next section.

In a second application of (23) and (24) , we make the identifications: ${\bf F}_{\texttt{1}}={\bf E}$ and ${\bf F}_{\texttt{2}}={\bf B}$ in (23) and (24) and subsequently use the Maxwell equations with material sources (SI units):

	$\displaystyle\nabla\cdot{\bf E}=\,\frac{1}{\epsilon_{0}}(\rho-\nabla\cdot{\bf P}),\qquad\qquad\;\nabla\cdot{\bf B}=0,$		(31)
	$\displaystyle\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial t}=0,\qquad\quad\nabla\times{\bf B}-\frac{1}{c^{2}}\frac{\partial{\bf E}}{\partial t}=\mu_{0}\bigg{(}{\bf J}+\nabla\times{\bf M}+\frac{\partial{\bf P}}{\partial t}\bigg{)},$		(32)

where ${\bf P}$ and ${\bf M}$ are the polarisation and magnetisation vectors, to get the retarded fields [12]:

	$\displaystyle{\bf E}=$	$\displaystyle\!-\nabla\bigg{(}{1\over 4\pi\epsilon_{0}}\int\!\frac{[\rho-\nabla^{\prime}\cdot{\bf P}]}{R}\,d^{3}r^{\prime}\bigg{)}-\frac{\partial}{\partial t}\bigg{(}\frac{\mu_{0}}{4\pi}\int\frac{[{\bf J}+\nabla^{\prime}\times{\bf M}+\partial{\bf P}/\partial t]}{R}d^{3}r^{\prime}\bigg{)},$
	$\displaystyle{\bf B}=$	$\displaystyle\nabla\times\bigg{(}\frac{\mu_{0}}{4\pi}\int\frac{[{\bf J}+\nabla^{\prime}\times{\bf M}+\partial{\bf P}/\partial t]}{R}\,d^{3}r^{\prime}\bigg{)},$			(34)

or more compactly,

$\displaystyle{\bf E}=-\nabla\Phi-\frac{\partial{\bf A}}{\partial t},\quad{\bf B}=\nabla\times{\bf A},$

(35)

where we have defined the retarded scalar and vector potentials as

$\displaystyle\Phi={1\over 4\pi\epsilon_{0}}\!\int\!\frac{[\rho-\nabla^{\prime}\cdot{\bf P}]}{R}\,d^{3}r^{\prime},\;{\bf A}=\frac{\mu_{0}}{4\pi}\!\int\!\frac{[{\bf J}+\nabla^{\prime}\times{\bf M}+\partial{\bf P}/\partial t]}{R}\,d^{3}r^{\prime}.$

(36)

Notice that if the vectors P and M are vanished in (33) and (34) then we recover (27) and (28).

In the last application we write ${\bf F}_{1}=c{\bf E}$ and ${\bf F}_{2}={\bf B}$ in (23) and (24), and subsequently use Maxwell’s equations with magnetic monopoles expressed in Gaussian units as

	$\displaystyle\nabla\cdot c{\bf E}=\,4\pi c\rho_{e},\qquad\qquad\;\nabla\cdot{\bf B}=4\pi\rho_{m},$		(37)
	$\displaystyle-\nabla\times c{\bf E}-\frac{\partial{\bf B}}{\partial t}=4\pi{\bf J}_{m},\quad\nabla\times{\bf B}-\frac{1}{c}\frac{\partial{\bf E}}{\partial t}=\frac{4\pi}{c}{\bf J}_{e},$		(38)

where $\rho_{e}$ and ${\bf J}_{e}$ are the electric charge and current densities and $\rho_{m}$ and ${\bf J}_{m}$ are the magnetic charge and current densities, to directly obtain the retarded fields with magnetic monopoles

	$\displaystyle{\bf E}=-\nabla\bigg{(}\int\!\frac{[\rho_{e}]}{R}\,d^{3}r^{\prime}\bigg{)}-\nabla\times\bigg{(}\int\frac{[{\bf J}_{m}]}{Rc}\,d^{3}r^{\prime}\bigg{)}-\frac{1}{c}\frac{\partial}{\partial t}\bigg{(}\int\frac{{\bf[}{\bf J}_{e}]}{Rc}\,d^{3}r^{\prime}\bigg{)},$		(39)
	$\displaystyle{\bf B}=-\nabla\bigg{(}\int\!\frac{[\rho_{m}]}{R}\,d^{3}r^{\prime}\bigg{)}+\nabla\times\bigg{(}\int\frac{[{\bf J}_{e}]}{Rc}\,d^{3}r^{\prime}\bigg{)}-\frac{1}{c}\frac{\partial}{\partial t}\bigg{(}\int\frac{[{\bf J}_{m}]}{Rc}\,d^{3}r^{\prime}\bigg{)},$		(40)

or more compactly:

$\displaystyle{\bf E}=-\nabla\Phi_{e}-\nabla\times{\bf A}_{m}-\frac{1}{c}\frac{\partial{\bf A}_{e}}{\partial t},\;\;\;{\bf B}=-\nabla\Phi_{m}+\nabla\times{\bf A}_{e}-\frac{1}{c}\frac{\partial{\bf A}_{m}}{\partial t},$

(41)

where we have defined the retarded electric and magnetic scalar potentials as

$\displaystyle\Phi_{e}=$

$\displaystyle\int\!\frac{[\rho_{e}]}{R}\,d^{3}r^{\prime},\;\;\Phi_{m}=\int\!\frac{[\rho_{m}}{R}\,d^{3}r^{\prime},$

(42)

and the retarded electric and magnetic vector potentials as

$\displaystyle{\bf A}_{e}=\int\frac{[{\bf J}_{e}]}{Rc}\,d^{3}r^{\prime},\;\;{\bf A}_{m}=\int\frac{[{\bf J}_{m}]}{Rc}\,d^{3}r^{\prime}.$

(43)

Notice that if the magnetic densities $\rho_{m}$ and ${\bf J}_{m}$ are vanished in (39) and (40) then we recover (27) and (28).

Standard textbook presentations of electromagnetism follow a different route to the electric and magnetic fields in the time-independent regime of Maxwell’s equations than in the time-dependent regime of these equations. While in the time-independent regime, the Helmholtz theorem of the vector analysis is commonly applied to find the electrostatic and magnetostatic fields in terms of their respective scalar and vector potentials, in the time-dependent regime Maxwell’s sourceless equations are commonly used to introduce the scalar and vector potentials which are then inserted into Maxwell’s source equations, obtaining two coupled second-order equations involving potentials, which are shown to be gauge invariant. The Lorenz-gauge condition is usually adopted to decouple these second-order equations and as a result, we arrive at two wave equations, which are solved to obtain the retarded scalar and vector potentials and by a subsequent differentiation of them we finally obtain the retarded electric and magnetic fields. It is evident that the usual method followed in the time-dependent regime is somewhat more complicated than the usual method used in the time-independent regime which is based in the Helmholtz theorem. It is clear that for pedagogical reasons it is worth formulating an extension of the Helmholtz theorem, which may be useful in the time-dependent regime of Maxwell’s equations. This has been made in the past and different extensions of the Helmholtz theorem to include the time-dependence of the fields have been formulated [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. But, as far as we are aware, none of these extensions of the Helmholtz theorem has yet been included in standard textbooks.

In our opinion, equations (23) and (24) for the retarded fields $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ could be included in a undergraduate presentation of Maxwell’s equations. Similarly, the analogous equation but for a single retarded field [18]:

$\displaystyle{\bf F}=-\nabla\int\!\frac{[\nabla^{\prime}\cdot{\bf F}]}{4\pi R}d^{3}r^{\prime}\!+\nabla\!\times\int\!\frac{[\!\nabla^{\prime}\!\times{\bf F}]}{4\pi R}d^{3}r^{\prime}\!+\frac{1}{c^{2}}\frac{\partial}{\partial t}\int\!\frac{[\partial{\bf F}/\partial t]}{4\pi R}d^{3}r^{\prime},$

(44)

could alternatively be included in such an undergraduate presentation. We notice that if any of the fields $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ defined by (23) and (24) is vanished then we obtain (44). The disadvantage of (23) and (24) is that they are somewhat complicated but their advantage is that they directly yield the retarded fields. In contrast, (44) has the advantage of being simpler but the disadvantage is that it yields the retarded fields in an indirect way.