#  Why the controversy over displacement currents never ends?

Masao Kitano

( Center for Quantum Information and Quantum Biology, Osaka University, Osaka
560-0043, Japan
Research Institule of Applied Science, Sakyoku, Kyoto 606-8202, Japan
Professor emeritus, Kyoto University, Kyoto 606-8501, Japan
)

Displacement current is the last piece of the puzzle of electromagnetic
theory. Its existence implies that electromagnetic disturbance can propagate
at the speed of light and finally it led to the discovery of Hertzian waves.
On the other hand, since magnetic fields can be calculated only with
conduction currents using Biot-Savart’s law, a popular belief that
displacement current does not produce magnetic fields has started to
circulate. But some people think if this is correct, what is the displacement
current introduced for. The controversy over the meaning of displacement
currents has been going on for more than hundred years. Such confusion is
caused by forgetting the fact that in the case of non-stationary currents,
neither magnetic fields created by conduction currents nor those created by
displacement currents can be defined. It is also forgotten that the effect of
displacement current is automatically incorporated in the magnetic field
calculated by Biot-Savart’s law. In this paper, mainly with the help of
Helmholtz decomposition, we would like to clarify the confusion surrounding
displacement currents and provide an opportunity to end the long standing
controversy.
Keywords: displacement current, Biot-Savart law, Ampere’s law, Maxwell-
Ampere’s law, Helmholtz’s decomposition, non-stationary current

##  1 Introduction

The time derivative of the electric flux density,
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$,
is named displacement current density, which is the final piece to complete
the hard puzzle of electromagnetic theory. This discovery made by James Clerk
Maxwell [1, 2] was possible only through his keen eyes forseeing its existence
from theoretical inevitability (1864).

He found the fact that the propagation velocity of the wave solution enabled
by the displacement current, was consistent with the speed of light, which was
already measured experimentally at that time. The value of constant
$(\mu_{0}\varepsilon_{0})^{-1}$ had been determined by Weber and Kohlrausch in
other context[3], where $\mu_{0}$ and $\varepsilon_{0}$ are the permeability
and permittivity of vacuum, respectively. Maxwell was convinced that light is
an electric and magnetic disturbance propagating in a vacuum. Later, H.R.
Hertz discovered radio waves (1888) in attempting to detect displacement
currents using a capacitor.

Displacement currents occupy an important position in electromagnetics.
However, owing to the fact that magnetic fields can be correctly calculated by
the Biot-Savart law, which does not seem to include the displacement current,
it has widely been claimed that the displacement current does not produce a
magnetic field. As a matter of fact, however, the Biot-Savart law implicitely
includes the contribution of displacement currents.

In this paper, we would like to clarify the confusion surrounding displacement
currents and its causes and help to promote correct understanding.

##  2 Magnetic Action of Electric Currents and Displacement Current

In 1820, Hans Christian Ørsted discovered that a compass needle swings in
response to an electric current flowing near it. That same year, the relation
between current and magnetic field was formulated in two ways; Biot-Savart’s
law and Ampère’s law, which correspond to Coulomb’s law and Gauss’s law in
electrostatics, respectively.

These magnetic field laws were based on the assumption that the current is
flowing through a closed circuit. Almost half a century later, considering the
case of unclosed current, as in the case of charging capacitor, Maxwell
theoretically derived the necessity of displacement currents.

His argument goes as follows. By taking the divergence of both sides of
Ampère’s equation,
$\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}$,
we have,
$0=\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}$,
with the identity $\mathop{\mathrm{div}}\mathop{\mathrm{curl}}=0$. In other
words, Ampère’s equation implicitely assumes divergence-free currents. This is
also called the ”steady-state current condition,” because the charge
conservation law,
$\partial_{t}\varrho=-\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}$,
implies steady charge distributions. ($\partial_{t}=\partial/\partial t$ is
used for brevity.) If this condition is not satisfied, i.e.,
$\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\neq
0$, Ampère’s law must be modified as follows:

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}.$
(1)

The displacement current density term,
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$
is added. Now, taking the divergence of both sides, we have

$\displaystyle
0=\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}=\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}\varrho.$

The time derivative of Gauss’s formula,
$\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}=\varrho$,
is used. This is consistent with the charge conservation law.

Based on this reasoning, in his treatise [4], Maxwell states

> One of the chief pecurialities of this treatise is the doctrine which it
> asserts, that the true electric current $\mathfrak{C}$ ($C$), that on which
> the electromagnetic phenomena depend, is not the same thing as
> $\mathfrak{K}$ ($J$), the current of conduction, but that the time variation
> of $\mathfrak{D}$ ($D$), the electric displacement, must be taken into
> account in estimating the total movement of electricity, so that we must
> write,
>
>
> $\displaystyle\mathfrak{C}=\mathfrak{K}+\dot{\mathfrak{D}},\quad\text{\footnotesize(Equation
> of True Currents)}.$

Hereafter we write the true (total) current as

$\displaystyle{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}.$

Equation (1) is now called Maxwell-Ampère’s equation. Maxwell’s
electromagnetic theory thus created is being organized by the followers and
spread to the academic world

Oddly, however, the doctrine that displacement currents do not create magnetic
fields began to circulate. The main reasons are

  * •

Even in the presence of displacement currents, magnetic field is calculated
correctly by the Biot-Savart equation.

  * •

A typical displacement current is one that occurs where the linear current is
interrupted. Then charges are accumulated at the endpoint and yield a
spherically symmetric electric field. The corresponding displacement current
is also spherically symmetric and the resultant magnetic field vanishes.

Various arguments against, for, or from a neutral standpoint about this
theory, some of which seem to deepen the confusion, are continuing in papers
and textbooks [5, 6, 7, 8, 9, 11, 12, 10, 13, 14, 15, 16, 17, 18].

In this paper, we will show the claim that displacement current does not
create a magnetic field is due to a lack of understanding of the mathematical
structure of electromagnetic fields. We mainly discuss from the following
points of view:

  * •

It is impossible in principle to separate magnetic fields into those caused by
”conduction currents” and those caused by ”displacement currents”.

  * •

The posed question ”does a displacement current create a magnetic field or
not?” is logically meaningless.

  * •

Contrary to popular perception, the Bio-Savart law perfectly includes the
effect of displacement currents implicitely [19].

##  3 Need for Displacement Current

In this section, we will reconfirm how the Ampère’s law is modified to account
for displacement currents.

The integral form of Ampère’s law,
$\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}$,
is

$\displaystyle\int_{C}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}=\int_{S}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}},$
(2)

where the closed path $C=\partial S$ is the edge of the surface $S$. In this
equality the surface $S$ can be arbitrary as long as the closed path $C$ is
its edge. In order for the integral to be the same regardless of the surface,
$\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}=0$
must be satisfied everywhere (steady-state current condition). Otherwise, for
surfaces $S_{1}\neq S_{2}$ with $\partial S_{1}=\partial S_{2}=C$, the
divergence theorem gives

$\displaystyle
0\neq\int_{V}\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\,{\mathrm{d}}v=\left(\int_{S_{1}}-\int_{S_{2}}\right){\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}},$

where $V$ is the volume enclosed by $S_{1}$ and $S_{2}$.

Figure 1: Displacement currents in charging capacitor

Consider a capacitor being charged with a constant current $I$, as shown in
Fig 1. We have two surfaces $S_{1}$ and $S_{2}$ that share the same circle $C$
encircling the capacitor as their respective circumferences. While the
hemisphere $S_{1}$ crosses the current $I$, the disk $S_{2}$ passes between
the electrodes of the capacitor and crosses no currents. The integrals of the
current density for these surfaces

$\displaystyle
I=\int_{S_{1}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}\quad\neq\quad\int_{S_{2}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}=0,$

are clearly not equal. But if we add the displacement current density
$\partial{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}/\partial
t$, then the surface integral for $S_{2}$ becomes

$\displaystyle\int_{S_{2}}\left({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\frac{\partial{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}}{\partial
t}\right)\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}=\int_{S_{2}}\frac{\partial{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}}{\partial
t}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}=\frac{{\mathrm{d}}}{{\mathrm{d}}t}Q(t)=I,$

and now the equality holds. The charge on the capacitor plate
$Q=\int_{S_{2}}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}$
can be derived from $D$ between the plates. With this model we can confirm
that Eq. (2) must be modified as

$\displaystyle\int_{C}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}=\int_{S}\left({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\frac{\partial{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}}{\partial
t}\right)\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}.$
(3)

This is the integral form of Maxwell-Ampère’s equation.

Figure 2: Counteracting displacement currents in current element connections

##  4 Inappropriate Problem Setting

In this section, we show that the question whether the displacement current
creates a magnetic field or not is totally meaningless.

The question can be broken down in the following way. When the total current
density
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$ is divided into the conduction current density $J$ and the displacement
current density
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$,
the magnetic field can be divided into two components as
${\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
c}}+{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
d}}$, correspondingly. And if we can prove
${\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
d}}=0\,(\neq 0)$, then the answer is no (yes). But, in the first place, what
are the (local) equations that these magnetic fields obey? The only possible
choice seems to be

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
c}}\stackrel{{\scriptstyle?}}{{=}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}},\quad\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
d}}\stackrel{{\scriptstyle?}}{{=}}\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}},$
(4)

but it leads us to the contradiction when we take the divergence;

$\displaystyle
0\stackrel{{\scriptstyle?}}{{=}}\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\neq
0,\quad
0\stackrel{{\scriptstyle?}}{{=}}\mathop{\mathrm{div}}\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}\neq
0.$

What did we introduce the displacement current for?

We have found that the magnetic field created the displacement current cannot
be defined and therefore, it makes no sense to ask whether such an undefinable
quantity is zero or not.

In general, if we want to solve the Maxwell-Ampère equation,
$\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$,
by superposition, then we should set

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{1}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{1}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{1},\quad\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{2}$
$\displaystyle={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{2}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{2},$
(5)

and divide not only the current density but also the displacement current
density term, so that each of them satisfies

$\displaystyle\mathop{\mathrm{div}}({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{1}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{1})=0,\quad\mathop{\mathrm{div}}({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{2}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{2})=0.$
(6)

This is the proper division of total current. On the other hand the division
of Eq. (4) makes no sense.

Let us consider a mathematical case to see why a simple-minded superposition
does not hold. For a general linear map $A:\,X\rightarrow Y$, from a space $X$
to another $Y$, let
$\text{Im}\,A=\\{A{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}\mid{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}\in
X\\}\subset Y,$ which is called the range or image of $A$. In order that the
linear equation

$\displaystyle
A{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}={\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}},$

has a solution
${\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}\in
X$, the condition
${\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}\in\text{Im}\,A$
must be met. Even when
${\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}={\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{1}+{\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{2}\in\text{Im}\,A$,
if
${\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{1},\;{\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{2}\notin\text{Im}A$
then the superposition cannot be used. Because

$\displaystyle
A{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}_{1}={\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{1},\quad
A{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}_{2}={\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{2},$

have no solutions. An example is

$\displaystyle A=\begin{bmatrix}1&1\\\
0&0\end{bmatrix},\quad{\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}=\begin{bmatrix}2\\\
0\end{bmatrix},\quad{\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{1}=\begin{bmatrix}1\\\
1\end{bmatrix},\quad{\mathchoice{\mbox{\boldmath$b$}}{\mbox{\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}{\mbox{\scriptsize\boldmath$b$}}}_{2}=\begin{bmatrix}1\\\
-1\end{bmatrix}.$

To make a situation where the solution is unique, we should set up another
linear equation,
$B{\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}{\mbox{\scriptsize\boldmath$x$}}}=0$,
which corresponds to
$\mathop{\mathrm{div}}(\mu_{0}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}})=0$
in our case.

##  5 Biot-Savart’s Law and Displacement Current

Here we will show that contrary to common belief Biot-Savart’s law include the
effect of displacement currents [21, 10].

For the sake of brevity, the Coulomb field is written as

$\displaystyle{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}):=\frac{{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}}{4\pi|{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}|^{3}}.$

With this, we have
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\cdot{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=\delta^{3}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
,
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=0$
,
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}(1/r)=-4\pi{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$,
where
$\delta^{3}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
is the delta function. (For calculation, we use
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\cdot$,
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\times$,
and $\nabla$, instead of $\mathop{\mathrm{div}}$, $\mathop{\mathrm{curl}}$,
and $\mathop{\mathrm{grad}}$.)

The electric flux density for a charge $q$ placed at the origin is
${\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{q}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=q{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
and that for an electric dipole
${\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}=q{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$
at the origin is

$\displaystyle{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{{\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=-({\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}\cdot{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}).$
(7)

For the current element
$I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$
placed at the origin, the magnetic field created at the point $r$ is given by
Biot-Savart law in the difference form

$\displaystyle\Delta{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}).$
(8)

The magnetic field due to the current $I$ flowing through the closed circuit
$L$ is given as a superposition (integral)

$\displaystyle{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=\oint_{L}{\mathrm{d}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})=\oint_{L}I{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}^{\prime}\\!\times\\!{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}),$
(9)

where
${\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}^{\prime}$
is a line element at position
${\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}$
on path $L$. Originally, the condition that ”$L$ is closed” was required by
Biot-Savart’s integrated formula initially.

Let us find the vortex of the magnetic field element (8). With the help of a
vector analysis formula, we have

$\displaystyle{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\times\Delta{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})={\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\times\left[(I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}})\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})\right]$
$\displaystyle\hskip
30.00005pt=I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}\left[{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\cdot{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})\right]-(I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}\cdot{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
$\displaystyle\hskip
30.00005pt=I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}\,\delta^{3}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})+{\frac{\partial}{\partial
t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{It\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})}=:\Delta{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}).$
(10)

We note that in addition to the original current element
$I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$
the additional term appears. This term is the time derivative of the electric
flux density (7) for the electric dipole
${\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}(t)=It\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$
at the origin. It means that if a constant current $I$ flows on the line
element
$\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$,
then there accumulates charge $\pm Q(t)=\pm It$ at each end
$\pm\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}/2$
to form an electric dipole.

The total current
$\Delta{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$ (10) satisfies the stationary condition and the magnetic field
$\Delta{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}$
is generated by this total current
$\Delta{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$.

The condition that $L$ is closed, which was originally assumed in the Biot-
Savart equation, is actually unnecessary. When a current is formally
integrated for an open path $L$ with start (end) point
${\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}_{1}$
(${\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}_{2}$),
we obtain

$\displaystyle{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
$\displaystyle=\int_{L}{\mathrm{d}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}\\!-\\!{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})$
$\displaystyle=I\int_{L}{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}^{\prime}\delta^{3}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}\\!-\\!{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})+\left.\rule{0.0pt}{10.76385pt}I{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}\\!-\\!{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})\right|_{{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}={\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}_{1}}^{{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}_{2}}.$
(11)

As shown in Fig. 2, when the current elements are connected (integrated), the
displacement currents from opposing endpoints cancel each other and only those
at the two extreme ends of the path remain. This is similar to the case where
small bar magnets are connected to form a long chain. The magnetic fields of
opposite polarity at the connection points cancel each other, and only the
magnetic fields from the poles at both ends survive.

In the end, the Biot-Savart’s equation (9) can be applied even for unsteady
(open) currents, just with the modification of integration path;
$\oint\rightarrow\int$, which is beyond the originally intended scope of
application.

Although it is an integral of the current distribution $J$, the total current
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
disp}}$ is automatically taken into account and the corresponding magnetic
field is given.

This fact has been pointed out by from time to time [19, 30], but many people
still mistakenly believe that when using Biot-Savart they can calculate the
magnetic field only due to the conduction currents that they give. But even if
you don’t order the displacment current, it will always come by itself. This
”hidden trick” is one of the sources of confusion over the displacement
current.

Even though both Biot-Savart’s law and Ampère’s law were established in the
same year (1920), only the former consealed the effect of displacement current
that will be exposed 45 years later.

Figure 3: Helmhortz decomposition of current distribution

.
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\hskip
20.00003pt=$ $\hskip
20.00003pt{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}\hskip 30.00005pt+$
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}$
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\hskip
20.00003pt=$ $\hskip
20.00003pt{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}\hskip 26.00009pt+$
$(-{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
disp}})$ Current element
${\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}\,\delta^{3}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
$({\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}\times{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}})\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
$({\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}\cdot{\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
Semi-infinite linear current .  Spherically symmetric current .  $0$ Charging
capacitor

Figure 4: Examples of Helmholtz decomposition of current distribution

##  6 Helmholtz Decomposition of Vector Fields

From a mathematical point of view, we take a closer look at how the Biot-
Savart equation automatically incorporates the effect of displacement
currents. The current density field $J$ (or a three dimensional vector field
in general) can be divided into the ”curl-free” component
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}$ and the ”divergence-free” component
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$, namely,

$\displaystyle{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})+{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}),$
$\displaystyle\hskip
10.00002pt\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=0,\quad\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=0.$
(12)

This is the so-called Helmholtz decomposition [19, 20, 22]. The uniqueness of
the decomposition requires that the field quantities converge quickly to zero
at infinity, which is satisfied in the present case. The curl-free and
divergence-free fields are also called the “longitudinal” and “transverse”
fields, respectively, which are denoted by subscripts “L” and “T”. The latter
names are derived from the relations of their Fourier transform to the
wavevector $k$;

$\displaystyle{\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}}\times(\mathcal{F}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}})({\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}})=0,\quad{\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}}\cdot(\mathcal{F}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}})({\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}{\mbox{\scriptsize\boldmath$k$}}})=0.$

The Biot-Savart law for the three-dimensional current density
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
is

$\displaystyle{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=\int_{V}\\!{\mathrm{d}}v^{\prime}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}),$
(13)

and its vortex is

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})-\int_{V}\\!{\mathrm{d}}v^{\prime}({\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}^{\prime}{\cdot}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}\\!-\\!{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}).$
(14)

The first term is just the given current density. With the charge conservation
of law
${\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}\cdot{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})+\partial_{t}\varrho({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}},t)=0$,
the second term turns out to be the displacement current density:

$\displaystyle\int_{V}\\!{\mathrm{d}}v^{\prime}\partial_{t}\varrho({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime},t){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})=\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}},t).$

We can verify that
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$
is longitudinal.

Equation (14) can be rewritten as

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})-\hat{L}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=(\hat{1}-\hat{L}){\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}),$

where $\hat{1}$ is the identity operator, and the operator $\hat{L}$ acts on
the vector field
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
to create a new vector field:

$\displaystyle(\hat{L}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}})({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
$\displaystyle:=\int_{V}\\!{\mathrm{d}}v^{\prime}({\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}{\mbox{\scriptsize\boldmath$\nabla$}}}^{\prime}{\cdot}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})){\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}).$
(15)

The operator $\hat{L}$ gives the longitudinal field components of a vector
field.

We define another operator $\hat{T}:=\hat{1}-\hat{L}$, which gives the
transverse field component
$\hat{T}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}$.
The operators $\hat{L}$ and $\hat{T}$ are equipped with the properties of
projection operators, namely,

$\displaystyle\hat{T}^{2}=\hat{T},\quad\hat{L}^{2}=\hat{L},\quad\hat{T}\hat{L}=\hat{L}\hat{T}=0.$

As shown in Fig. 3, the Biot-Savart law gives the magnetic field due to the
transverse component of the current density
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}=\hat{T}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}-{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}$, or due to the total current density

$\displaystyle{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}:={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
disp}}.$

From these equations, we know that the displacement current is the curl-free
(longitudinal) component with sign changed of the given current:

$\displaystyle{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
disp}}=-{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}\>(=\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}).$
(16)

Adding the displacement current density
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
disp}}$ to the current density $J$ means the cancellation of the longitudinal
component
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}$ to yield
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$ or
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$. The difference between the subtraction and the sum gives a very
different impression.

In the cource of deriving the Biot-Savart’s law from Maxwell’s equations, we
confirm why the former includes the effect of displacement currents. With the
curl of the Maxwell-Ampères’s law,
$\mathop{\mathrm{curl}}\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$,
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\,+\,\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$,
and
$\mathop{\mathrm{div}}(\mu_{0}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}})=0$,
we have

$\displaystyle\nabla^{2}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=-\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}},$

where
$\mathop{\mathrm{curl}}\mathop{\mathrm{curl}}=\mathop{\mathrm{grad}}\mathop{\mathrm{div}}-\nabla^{2}$
is used. The solution to this (vector) Poisson’s equation [22] is

$\displaystyle{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=\int{\mathrm{d}}v^{\prime}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})=:(\hat{f}_{\mbox{\scriptsize
T}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}})({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}),$
(17)

where the operator $\hat{f}_{\mbox{\scriptsize T}}$ is defined so as to map a
transverse current
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$ to the corresponding magnetic field $H$. This map, which can be written
symbolically
${\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=\mathop{\mathrm{curl}}^{-1}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$, is invertible, i.e., one-to-one. For a general current $J$, we should
have

$\displaystyle{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=\hat{f}_{\mbox{\scriptsize
T}}\hat{T}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}=:\hat{f}_{\mbox{\scriptsize
BS}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}},$

which correponds to the Biot-Savart’s law (13). The two-step operation,
$\hat{f}_{\mbox{\scriptsize BS}}=\hat{f}_{\mbox{\scriptsize T}}\hat{L}$ is not
invertible. The current distribution cannot be uniquely determined from the
magnetic field. This fact is overlooked so often.

In order to fix the problem of improper division of Eq. (4), we can apply
$\hat{T}$ for each current.

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=\hat{T}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}},\quad\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
d}}=\hat{T}(\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}})=0.$

Then, in the second equation, the displacement current
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}$
certainly disappears and
${\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
d}}$ vanishes. But we should remember that it reapears in the first equation
and contribute to $H$. We cannot erase the displacement current.

##  7 Example of Helmholtz Decomposition of Current Distribution

In Fig.4, the Helmholtz decomposition is shown for several current
distributions. All of these examples has been used to discuss displacement
currents. Each of them is briefly described below. We have already mentioned
in Sec. 5 about the current elements in the first row. In the case of point
charge $q$ moving at velocity $v$, we can set
${\mathchoice{\mbox{\boldmath$p$}}{\mbox{\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}{\mbox{\scriptsize\boldmath$p$}}}=q{\mathchoice{\mbox{\boldmath$v$}}{\mbox{\boldmath$v$}}{\mbox{\scriptsize\boldmath$v$}}{\mbox{\scriptsize\boldmath$v$}}}$,
instead of
$I\Delta{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}$.

###  7.1 Semi-infinite linear current

If a constant current $I$ is flowing along the half line ($z\leq 0$) along the
$z$-axis, the charge must be accumulated at the origin as $Q(t)=It+Q(0)$,
where $Q(0)$ is the charge at time $t=0$.

Electric flux density of Coulomb type and the associated displacement current
density
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}(t,{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=I{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})$
are generated. We can derive the magnetic fields in two ways, i.e., with Biot-
Savart’s law and Maxwell-Ampère’s formula.

Using cylindrical coordinates $(\rho,\phi,z)$, the Biot-Savart law for an open
path,
$L=\\{{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime}=z{\mathchoice{\mbox{\boldmath$e$}}{\mbox{\boldmath$e$}}{\mbox{\scriptsize\boldmath$e$}}{\mbox{\scriptsize\boldmath$e$}}}_{z}\mid-\infty<z\leq
0\\}$, we have

$\displaystyle
H_{\phi}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}})=H_{\phi}(\rho,z)=\int_{L}I{\mathrm{d}}z{\mathchoice{\mbox{\boldmath$e$}}{\mbox{\boldmath$e$}}{\mbox{\scriptsize\boldmath$e$}}{\mbox{\scriptsize\boldmath$e$}}}_{z}\times{\mathchoice{\mbox{\boldmath$G$}}{\mbox{\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}{\mbox{\scriptsize\boldmath$G$}}}({\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}-{\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}{\mbox{\scriptsize\boldmath$r$}}}^{\prime})$
$\displaystyle=\frac{I}{4\pi}\int_{-\infty}^{0}\frac{\rho\,{\mathrm{d}}z^{\prime}}{[(z-z^{\prime})^{2}+\rho^{2}]^{3/2}}=\frac{I}{4\pi\rho}\left(1-\frac{z}{\sqrt{z^{2}+\rho^{2}}}\right).$

Secondly, to apply Maxwell-Ampère’s formula, consider a spherical surface with
radius $R=\sqrt{\varrho^{2}+z^{2}}$ centered at the origin. We have a latitude
line $C$ defined by $\theta=\tan^{-1}(\varrho/z)=\text{const}$. Let $S_{+}$
($S_{-}$) be the northern (southern) spherical crown cut by $C$. Note that
$C=\partial S_{+}=-\partial S_{-}$. Applying the Maxwell-Ampère formula (1)
for both surfaces, we have

$\displaystyle\oint_{C}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}=\int_{S_{+}}\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}=-\int_{S_{-}}({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}})\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}.$

The left hand side is $2\pi\rho H_{\phi}(\rho,z)$. The middle side is
evaluated as follows. Since the magnitude of the displacement current on a
sphere of radius $R$ is $\partial_{t}D=I/(4\pi R^{2})$, it is perpendicular to
the sphere, and the area of $S_{+}$ is $|S_{+}|=2\pi R^{2}(1-\cos\theta)$, we
have

$\displaystyle\partial_{t}D|S_{+}|=\frac{I}{2}(1-\cos\theta).$

Similary for the right hand side using $|S_{-}|=2\pi R^{2}(1+\cos\theta)$ and
$\int_{S_{-}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\cdot{\mathrm{d}}{\mathchoice{\mbox{\boldmath$S$}}{\mbox{\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}{\mbox{\scriptsize\boldmath$S$}}}=-I$,
we have

$\displaystyle I-\partial_{t}D|S_{-}|=I-\frac{I}{2}(1+\cos\theta).$

Thus, the same result is obtained by using either of the surfaces:

$\displaystyle
H_{\phi}(\rho,z)=\frac{I}{4\pi\rho}(1-\cos\theta)=\frac{I}{4\pi\rho}\left(1-\frac{z}{\sqrt{\rho^{2}+z^{2}}}\right).$

This result agrees with the previous result by Bio-Savart’s law. In this
method we had to consider the displacement currents explicitly, otherwise, the
solution is not uniquely determined.

###  7.2 Spherically symmetric current distribution

To back up the false claim that displacement currents do not create magnetic
fields, the fallacious theory that ”spherically symmetric currents do not
create a magnetic field due to their symmetry” is developed. Combining Biot-
Savart’s equation and symmetry for each part of the current, it is attempted
to to show that the magnetic field is zero.

As an example, let’s take a look at section 9.2 of Purcell’s textbook [18].
Noting that a curl-free field like displacement current density can be written
as a superposition of spherically symmetric Coulomb-type vector fields, it
continues,

> … the magnetic fields of any radial, symmetrical current distribution,
> calculated via Biot-Savart, is zero. To understand why, consider a radial
> line through a given line through a given location. At this location, the
> Biot-Savart’s contributions from a pair of points symmetrically located with
> respect to this line are equal and opposite, as you can verify. The
> contributions therefore cancel in pairs, yielding zero field at the given
> location.

Here the fact is forgotten that Biot-Savart’s law includes the effect of
displacement current, which in this case is spherically symmetric but flows in
the opposite direction.

In his famous series of text book [23], Feynman correctly explained the
situation of spherically symmetric current using a model where a small sphere
with radioactive material is squirting out some charged particles.

Physically speaking, as shown in Fig. 3, spherically symmetric currents can be
thought of as a large number of semi-linear currents isotropically combined at
a single point. In this case, the displacement currents at the endpoints add
up to exactly cancel the original currents. The total current becomes zero in
all places, and therefore the magnetic field is also zero. The magnetic field
vanishes not owing to the cancellation of real currents. In short, if the
symmetric current (conducting or displacement) is purely longitudinal, the
total current is zero.

Figure 5: Charging capacitor. The spacing $d$ is exaggerated compared with the
radius $R$. The displacement current is uniformly distributed.

###  7.3 Charging capacitor

The Charging capacitor problem is one of the reasons why Maxwell came up with
the idea of displacement current. The debate continues over this model, as to
whether the displacement current between the electrodes creates a magnetic
field or not [14, 15, 24].

Surprisingly, miscalculations are found, from time to time, in dealing with
this problem. For example, Roche [14] made wrong calculation in his historical
survey paper on the controversy over the reality of displacement current.
Jackson[15] pointed out the mistake and complained that the paper is marred by
imprecision, sloppy notation, and downright mistakes.

For simplicity, let us assume an axisymmetric system around $z$-axis as shown
in Fig. 5. Let $R$ be the radius of the electrodes and $d$ be the spacing. The
system is assumed to be charged by a constant current $I$ through the
conducting wires. Assuming that $d\ll R$, the electric flux density between
the electrodes is spatially uniform and given as $D_{z}(t)=It/\pi R^{2}$.

Since the current has no angular component, $J_{\phi}=0$, the magnetic field
has only the azimuthal component $H_{\phi}$. The magnetic field due to the
current $I$ on the straight conductor is

$\displaystyle H^{\mbox{\scriptsize(1)}}_{\phi}(\rho,z)=\frac{I}{2\pi\rho}.$
(18)

The current $I$ in the wire $z<-d/2$ reaches the eletrode at $z=-d/2$ and
spreads radially and the electrode is charged with a uniform charge surface
density. The linear density of this radial current $K_{\rho}(\rho)$ is
dependent on $\rho$ and can be determind by the charge conservation condition
and the uniformity. With the two-dimensional divergence formula we have
$\rho^{-1}({\mathrm{d}}/{\mathrm{d}}\rho)[\rho K_{\rho}(\rho)]=\text{const}$.
Under the boundary conditions $(2\pi\rho K_{\rho})(0)=I$ and $K_{\rho}(R)=0$,
the differential equation can be solved;

$\displaystyle
K_{\rho}(\rho)=\frac{I}{2\pi\rho}\left(1-\frac{\rho^{2}}{R^{2}}\right).$ (19)

For the other electrode at $z=d/2$, the surface current density is
$-K_{\rho}(\rho)$.

The straight conductor has a gap $-d/2<z<d/2$ of length $d$. The
$H^{\mbox{\scriptsize(1)}}_{\phi}$ includes its contribution. Therefore, we
have to introduce a current of $-I$ flowing in this segment [15]. (Roche [14]
missed this contribution, which remains even in the limit, $d\rightarrow 0$.)
If we connect this current element to the two surface currents, the
displacement currents from these two connection points become zero and only
those between the electrodes remain.

In order to apply Ampère-Maxwell’s law, we set up a circular loop,
$\rho=\text{const}$ between the electrodes ($-d/2<z<d/2$). For $\rho\leq R$,
we have $2\pi\rho H^{(2)}_{\phi}=-I+I{\rho^{2}}/{R^{2}}$, or

$\displaystyle
H^{(2)}_{\phi}(\rho,z)=\frac{I}{2\pi\rho}\left(-1+\frac{\rho^{2}}{R^{2}}\right),$

For $\rho>R$, we note $H^{(2)}_{\phi}=0$. By superposition, we have the
magnetic field of all position as

$\displaystyle H_{\phi}(\rho,z)=H^{(1)}_{\phi}(\rho,z)+H^{(2)}_{\phi}(\rho,z)$
$\displaystyle\hskip 12.91663pt=\begin{cases}\displaystyle\hskip
5.0pt\frac{I}{2\pi}\frac{\rho}{R^{2}}&(\rho\leq R,\,-d/2<z<d/2)\\\\[8.61108pt]
\displaystyle\hskip
5.0pt\frac{I}{2\pi}\frac{1}{\rho}&(\text{otherwise})\end{cases}.$ (20)

The difference between the magnetic fields inside and outside the electrodes
is equal to the surface current density (19):

$\displaystyle H_{\phi}(\rho,d/2+0)-H_{\phi}(\rho,d/2-0)=K_{\rho}(\rho).$

This current splitting in Fig. 5, which follows the rule (5), is a clever way
to avoid displacement currents except those between these electrodes.

Many experiments on charging capacitor have been conducted [25, 26, 27, 28].
Especially, the careful measurement by Bartlett and Corle [27] seems very
precise and consistent with Eq. (20). But the title of paper “Measuring
Maxwell’s displacement current inside a capacitor” is misleading, because what
was measured was the magnetic field inside the capacitor. As authors admitted
in the very last paragraph (and later in [29]),

> What we have shown, then, is that the Biot-Savart law applies to open as
> well as to closed circuits. One may write the differential form of this law
> as
> ${\mathrm{d}}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}=I{\mathrm{d}}{\mathchoice{\mbox{\boldmath$l$}}{\mbox{\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}{\mbox{\scriptsize\boldmath$l$}}}\times{\mathchoice{\mbox{\boldmath$R$}}{\mbox{\boldmath$R$}}{\mbox{\scriptsize\boldmath$R$}}{\mbox{\scriptsize\boldmath$R$}}}/R^{3}$,
> without the usual caveat that only the integral around a closed loop is
> meaningful.

The experiment does not answer the question whether the displacement current
generate magnetic field or not (or the question itself is meaningless). In my
opinion, it should be stressed that the vortex of the magnetic field between
the electrodes was actually measured. The existence of such vortices cannot be
explained without displacement currents. Further more it might be interesting
to demonstrate that when the wiring path to the capacitor is changed (e.g., to
form a coil) so as the magnetic field between the electrodes is disturbed, the
uniform vortex is still maintained.

Magnetic fields generated by a capacitor discharging through the partially
conducting spacer are dicussed in confusion from time to time (leaky
capacitor) [13]. The field profile can better be understood in terms of that
for a current element (see Fig. 4).

##  8 Displacement Currents and Electromagnetic Waves

Let us look at the relationship between displacement currents and
electromagnetic waves utilizing the Helmholtz decomposition (12). From two of
the Maxwell equations and the constitutive relation of vacuum

$\displaystyle\frac{\partial{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}}{\partial
t}=-\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}},\quad\frac{\partial{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}}{\partial
t}=\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}$
(21)
$\displaystyle{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}=\varepsilon_{0}{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}},\quad{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}=\mu_{0}^{-1}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}},$

we have the equations for the plane waves propageted in the $z$ direction
(assuming $\partial_{x}=\partial_{y}=0$ and $x$ polarization, i.e., $E_{y}=0$)

$\displaystyle\varepsilon_{0}\frac{\partial E_{x}}{\partial z}=-\frac{\partial
H_{y}}{\partial t},\quad{\mu_{0}\frac{\partial H_{y}}{\partial
z}}=-\frac{\partial E_{x}}{\partial t}.$

The d’Alembert solution to the hyperbolic partial differential equations is

$\displaystyle E_{x}(t,z)=f(z-c_{0}t)+g(z+c_{0}t).$

The $f$ and $g$ are arbitrary functions. These waves are propagated at the
velocity $\pm c_{0}=\pm\sqrt{\mu_{0}/\varepsilon_{0}}$. If there were no
displacement current term, then $c_{0}\rightarrow\infty$ and no wave solution
existed.

So far, we assumed the case where the steady current condition is violated
($\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}\neq
0$, i.e.
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}\neq
0$). We now relax the condition further and consider the case where the
magnetic field is also time-varying
($\partial_{t}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}\neq
0$).

We separate Maxwell’s equations into the longitudinal and transverse
components. Since the magnetic flux density satisfies
$\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}=0$,
we have
${\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}\equiv{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
T}}$, or
${\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
L}}=0$. The equation of electromagnetic induction is

$\displaystyle\mathop{\mathrm{curl}}({\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}}_{\mbox{\scriptsize
T}}+{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}}_{\mbox{\scriptsize
L}})=\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}}_{\mbox{\scriptsize
T}}=-\partial_{t}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
T}}.$ (22)

For the electric flux density, from
$\mathop{\mathrm{div}}({\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}}+{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}})=\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}=\varrho$, the longitudinal component
${\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}$ is related to the charge density $\varrho$. On the other hand, the
transverse component
${\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}}=\varepsilon_{0}{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}}_{\mbox{\scriptsize
T}}$ is related to the electromagnetic induction equation (22).

Substituting
${\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}={\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
T}}=\mu_{0}^{-1}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
T}}$, into Maxwell-Ampère’s equation, we have

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
T}}$
$\displaystyle={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}}+({\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}).$

Since the left-hand side is a transverse field (divergence-free), as in the
case of a stationary field, the longitudinal components cancel each other;
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}\equiv 0$. This can be regarded as the role of the longitudinal component
of the displacement current. In summary, we have

$\displaystyle\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$E$}}{\mbox{\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}{\mbox{\scriptsize\boldmath$E$}}}_{\mbox{\scriptsize
T}}=-\partial_{t}{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
T}},\quad\mathop{\mathrm{curl}}{\mathchoice{\mbox{\boldmath$H$}}{\mbox{\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}{\mbox{\scriptsize\boldmath$H$}}}_{\mbox{\scriptsize
T}}={\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}},$ (23)
$\displaystyle{\mathchoice{\mbox{\boldmath$B$}}{\mbox{\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}{\mbox{\scriptsize\boldmath$B$}}}_{\mbox{\scriptsize
L}}=0,\quad\mathop{\mathrm{div}}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}=\varrho,\quad{\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
L}}+\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
L}}=0.$

Since there is no divergence for both fields
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}$ and
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}}$, they can be splitted properly and the magnetic field created by each of
them can be defined. By the coupling of the first two equations due to the
transverse displacement currents, hybridization of the transverse components
of the electric and magnetic fields, i.e., electromagnetic wave modes are
enabled.

In particular, for
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
T}}=0$, a free solution exists. Since the transverse displacement current
$\partial_{t}{\mathchoice{\mbox{\boldmath$D$}}{\mbox{\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}{\mbox{\scriptsize\boldmath$D$}}}_{\mbox{\scriptsize
T}}$ is not bounded by the real current, the electromagnetic wave can
propagate far from the source. In fact the dependence of the longitudinal
component of the displacement current on the distance from the source is at
most $1/R^{2}$, that of the electromagnetic wave, i.e., the transverse
component, is $1/R$.

Sometimes the Jeffimenko equation, which is a dynamical extension of
Coulombs’s law and Biot-Savart’s law, is used to rule out the contribution of
displacement currents in quasi-static cases [31]. But arguments in this
direction only complicate things and do not seem useful.

##  9 Summary

The controversy over the meaning of displacement currents tends to get lost in
the choice between creating a magnetic field or not. Such confusion arises
from the following facts.

  * •

In the case of non-stationary currents, neither magnetic field created by
conduction current nor that created by displacement current can be defined.

  * •

In solving Maxwell-Ampère’s equation by superposition, the right-hand side
cannot be divided arbitrary ignoring the inseparability of the displacement
current and the current.

  * •

The effect of displacement current is automatically incorporated in the
magnetic field calculated by Biot-Savart’s law.

The existence of displacement current is subtle and elusive, and it was only
discovered through Maxwell’s deep insight. He emphasized the importance of the
unity of the current and the displacement current, and defined the sum of them
as total current
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$. He wrote down the fundamental equations of the electromagnetic field
using the total current
${\mathchoice{\mbox{\boldmath$J$}}{\mbox{\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}{\mbox{\scriptsize\boldmath$J$}}}_{\mbox{\scriptsize
tot}}$.

As we have seen, each of the longitudinal and transverse components of
displacement currents plays a different role. Although the former was the
initial impetus for the introduction of displacement currents, it plays a
shadow role to counteract the longitudinal component of the current. On the
other hand, the latter plays a prominent role in propagating electromagnetic
waves

While attempting to distinguish the roles of current and displacement current,
researchers have fallen for the trap of Bio-Savart’s equation. In particular,
the statement that the displacement current does not produce a magnetic field
may leads to underestimation of displacement currents, or even to lost sight
of its essential role in electromagnetic waves.

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