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Visualizing dipole radiation
To cite this article: Raimund V Girwidz 2016 Eur. J. Phys. 37 065206

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European Journal of Physics
Eur. J. Phys. 37 (2016) 065206 (13pp)

doi:10.1088/0143-0807/37/6/065206

Visualizing dipole radiation
Raimund V Girwidz
Didaktik der Physik, Ludwig-Maximilians-Universität München, Theresienstr. 37,
80333 München, Germany
E-mail: girwidz@lmu.deReceived 1 August 2016, revised 8 September 2016
Accepted for publication 21 September 2016
Published 17 October 2016
Abstract

The Hertzian dipole is fundamental to the understanding of dipole radiation. It
provides basic insights into the genesis of electromagnetic waves and lays thegroundwork for an understanding of half-wave antennae and other types.
Equations for the electric and magnetic ﬁelds of such a dipole can be derived
mathematically. However these are very abstract descriptions. Interpretingthese equations and understanding travelling electromagnetic waves are highly
limited in that sense. Visualizations can be a valuable supplement that vividly
present properties of electromagnetic ﬁelds and their propagation. The
computer simulation presented below provides additional instructive illustrations for university lectures on electrodynamics, broadening the experience
well beyond what is possible with abstract equations. This paper refers to a
multimedia program for PCs, tablets and smartphones, and introduces anddiscusses several animated illustrations. Special features of multiple representations and combined illustrations will be used to provide insight into
spatial and temporal characteristics of ﬁeld distributions—which also draw
attention to the ﬂow of energy. These visualizations offer additional information, including the relationships between different representations that
promote deeper understanding. Finally, some aspects are also illustrated that
often remain unclear in lectures.
Keywords: dipole radiation, Hertzian dipole, half-wave dipole, multimedia
visualization, animations, illustrations(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
The dipole antenna is the simplest and most widely used type of antenna in radio andtelecommunications. Understanding the Hertzian dipole is the ﬁrst step toward understanding
0143-0807/16/065206+13$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

Eur. J. Phys. 37 (2016) 065206

R V Girwidz

half-wave antennae and other types. The so called ‘ideal dipole’ was discussed in 1888 by
Heinrich Hertz as a part of his pioneering investigations into radio waves.
The oscillating electric dipole is a central wave-producing mechanism in electromagnetism. It is a standard topic for lectures in physics, since it points to the fundamental
mechanism that is responsible for electromagnetic radiation and waves. However, deriving
the equations for the electric and magnetic ﬁelds is difﬁcult. Moreover, it is not easy to
explain the spatial and time-dependent behavior to physics students, in that way. Nevertheless, dipole radiation is an important topic for electrodynamic courses in mathematical andtheoretical physics [1–3], as well as for bachelor university students during their ﬁrst years inexperimental physics [4, 5].
It can be quite instructive (and impressive) to visualize the time-dependent electric andmagnetic ﬁelds that are created by a radiating dipole, and the development of electromagnetic
waves. Suitable illustrations and animations are useful. The computer simulations presented
here can provide additional instructive illustrations for university lectures on electrodynamics
that broaden the experience well beyond what is possible with abstract equations.
The intention of this article and the corresponding computer application is to help invisualizing the characteristics of dipole radiation. The following text describes ﬁve speciﬁc
illustrations focusing on various aspects (as well as combinations thereof). Corresponding
animations can be created by the HTML5-applet ‘DipoleRadiation’. This app was written
speciﬁcally to illustrate dipole radiation, and can even be run on smartphones. Thus, it can beused in lectures as well as for individual studies.
Special topics for the illustrations are:• radiation pattern and structure of the electromagnetic ﬁeld;• detachment of ﬁeld lines and the generation of closed loops;• ﬁeld lines of E and H in three dimensional pictures and animations;• near ﬁeld and far ﬁeld, phase relationship between E and H;• ﬂow of energy, illustrations with Poynting vectors;• half wave dipole in contrast to a Hertzian dipole;• electric ﬁeld lines seen as contour lines of a potential function.
For devices with low computing power, a selection of pre-compiled video clips is alsoavailable.
There are many resources on the Internet that deal with dipole radiation. However, highresolution illustrations with ﬁeld lines and scalable sections that permit differentiation
between near ﬁeld and far ﬁeld are rare. Furthermore, combined illustrations, e.g. combinations of ﬁeld lines and Poynting vectors to show the ﬂow of energy, are hard to ﬁnd. Often
simulations are written in programming languages for speciﬁc operating systems, and are notavailable in HTML5. Thus, in most cases only animations (without the option to change
parameter values) can be run on arbitrary devices. A selection of additional online resources
with animations, which are helpful for lectures, can be found in [10–13].
2. Methods
2.1. Illustrations and multiple representations

Images can highlight different features of electromagnetism from what is captured bymathematical equations and can illustrate some of the unique characteristics of ﬁelds moreclearly. Additionally, animations provide speciﬁc valuable insights into time-dependent
processes. Moreover, beside a spatial and temporal overview, the combination of graphical2Eur. J. Phys. 37 (2016) 065206

R V Girwidz

Figure 1. Snapshot showing electric ﬁeld lines of a Hertzian dipole in combination with
the ﬂow of energy.

tools can focus on relations between different aspects, e.g. between a ﬁeld distribution and its
corresponding ﬂow of energy (see ﬁgure 1 for an example).
From the perspective of multimedia theory, Ainsworth in 1999 identiﬁed various functions of computer illustrations for learning and understanding [6]. Multiple representations
can ‘provide complementary information’, ‘constrain possible (mis)interpretations’ that arise
from one illustration in the use of another, and, ﬁnally, help learners to ‘construct a deeper
understanding’ of a phenomenon [6]. Understanding can be aided by promoting abstraction
from speciﬁc representations (which encourages generalization), and by teaching the relationships between different representations. Thus, various representations and also combinations of several images should be offered. Multiple visualizations help to highlight and tointerrelate different aspects of the electromagnetic ﬁeld formed by an oscillating electric
dipole.
2.2. Dipole radiation (Hertzian dipole)

The Hertzian dipole is more of a theoretical construction than a physical antenna. It has an
inﬁnitesimal length, which means that the wavelength of the emitted electromagnetic radiation is much larger than the length of the antenna. The electromagnetic radiation from a
Hertzian dipole can be calculated precisely by mathematical means. From this point, thesuperposition principle permits analytical or numerical calculation of the radiation from more
complex antennae, based on the solution for the ﬁelds from a Hertzian dipole. Hence,
becoming familiar with fundamental characteristics of the Hertzian dipole radiation is aﬁrst step.
Electric dipoles are characterized by their dipole moment, a vector quantity. Here, we
examine an oscillating electric dipole, with a maximum dipole moment p0 along the ydirection and a harmonic angular frequency ω. It can be described by the following timedependent electric dipole momentp (t ) = p0 ⋅ yˆ ⋅ sin wt .
( 1)
As the dipole moment varies sinusoidally in magnitude, the electric ﬁeld produced by the
dipole also varies in magnitude. Furthermore, because the electric current changes in time, the3Eur. J. Phys. 37 (2016) 065206

R V Girwidz

magnetic ﬁeld produced by the current varies in magnitude and direction. However, changes
in the electric and magnetic ﬁelds do not come about everywhere instantaneously.
Oscillations in these ﬁelds move away from the dipole at the speed of light (at least in thefar ﬁeld, see below) and with diminishing amplitude. Thus, the electric and magnetic ﬁelds
depend on position and time.
In a vacuum, the ﬁeld produced by this oscillating dipole can be derived exactly by using
the retarded potential formulation (see textbooks on electrodynamics).
The results for theelectric ﬁeld vector E and the magnetic ﬁeld vector H can be described by the following
equations (2) and (3)


  
  
  


p̈
p
3 ( p ⋅ r ) ⋅ r
3 (r ⋅ p ) ⋅ rp⎞( p̈ ⋅ r ) ⋅ r
1 ⎛
+
+
+
(2 )
E=
⎜- 2
⎟
4pe0 ⎝ c ⋅ r
c ⋅ r2c ⋅ r4c2 ⋅ r 3
r5
r3 ⎠






p ´ r ⎞1 ⎛ p̈ ´ r
H=
+
⎜
⎟;
4p ⎝ c ⋅ r 2
r3 ⎠

r⎞

⎛
with p = p ⎜t - ⎟ .
⎝
c⎠

(3)

Equations (2) and (3) are taken from [3, p 432].
Here, p represents the electric dipole moment, p and p ̈ are the ﬁrst and the second
derivative of this quantity with respect to time, e0 is the electric permittivity of free space, c is
the speed of light and r is the distance from the dipole.
2.3. Methods for calculating field lines

Electric ﬁeld lines can be calculated using Euler’s method, which can be understood even by
freshmen: from an arbitrary starting point, a nearly inﬁnitesimally short line is drawn in the
direction of the ﬁeld vector (according to equation (1)). From the position at the end of theline, the local ﬁeld vector is computed anew, and again a very short line segment is drawn.
This procedure is repeated until the starting point is reached again or the line ends up at theposition of the dipole.
However, even when using very short line segments, this procedure will lead to systematic deviations. Therefore, a sort of Runge–Kutta-procedure is used here. Instead of one
calculation for the direction of the ﬁeld line at a certain point, four estimates of the slope (i.e.
the calculation of the Ex/E and Ey/E components) are combined to gain more accuracy.
The ﬁrst direction is given by Ex1/E and Ey1/E at the initial position (x0, y0, in Cartesian
coordinates).
The next slope (Ex2/E and Ey2/E) results from a calculation of E at the position(x 0 + Ex1 E ⋅ d 2 , y0 + Ey1 E ⋅ d 2), where d is a very short distance (if converted into
display coordinates it is even smaller than one pixel).
The third slope (Ex3/E and Ey3/E) comes from the ﬁeld at the position(x 0 + Ex 2 E ⋅ d 2 , y0 + Ey2 E ⋅ d 2), and the fourth slope (Ex4/E and Ey4/E)
from (x 0 + Ex3 E ⋅ d , y0 + Ey3 E ⋅ d ).
Then a ‘weighted sum’ of these slopes is used to get the ﬁnal estimate for the next point
the ﬁeld line runs throughx = x 0 + (Ex1 6 + Ex 2 3 + Ex3 3 + Ex 4 6) E ⋅ d ; and( 4)y = y0 + (E y1 6 + E y2 3 + E y3 3 + E y 4 6) E ⋅ d .

(5)

The distance d is so small that the accuracy of the ﬁeld lines is better than one pixel .
1

1

One graphical test for this is to create ﬁeld lines by starting from different positions and stepping into the opposite
direction.
4Eur. J. Phys. 37 (2016) 065206

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Figure 2. Snapshot showing electric ﬁeld lines of a Hertzian dipole.

Figure 3. Snapshot showing electric ﬁeld lines of a Hertzian dipole, focusing at thetransition between near ﬁeld and far ﬁeld.

A completely different technique, which was also used by Heinrich Hertz, calculates apotential function where the ﬁeld lines appear as ‘contour lines’. A corresponding image isshown and explained further below.

3. Results and application
3.1. Visualizing the field structure and radiation pattern

Common images of ﬁeld lines are well known from various textbooks. A local electric ﬁeld
vector is always tangent to these ﬁeld lines. In ﬁgures 2 and 3, the directions of the ﬁeld lines
are indicated by arrows, and different phases of the ﬁeld are color-coded, indicating different
directions of circulation. The animations show electric ﬁeld lines due to a vertically oscillating
electric dipole. In this article, only snapshots from a movie clip can be shown, but the5

Eur. J. Phys. 37 (2016) 065206

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Figure 4. Electric ﬁeld lines (blue and green) and magnetic ﬁeld lines (red) at rightangles, composing the electromagnetic wave radiated by the oscillating dipole.

animations show time-dependent spreading and progression of ﬁeld properties more clearly
than a series of pictures. When the oscillating dipole vector is at its maximum, the ﬁeld lines
next to the dipole are nearly those of a static dipole, emerging from a positive charge andending up at a negative charge. However, at distances on the order of half a wavelength or
greater, the ﬁeld lines are completely detached from the dipole, characterizing a radiation ﬁeld
which propagates freely (without being attached to charges) in a vacuum at speed c.
The animation can be viewed either frame-by-frame with adaptable time intervals or byrepeating in a loop.
3.2. 3D-view for E- and H-lines

Since we are examining a three-dimensional process, it is helpful to show three-dimensional
representations as well. A spatial impression can be useful for future applications—such as
the superposition of the radiation from two or more dipoles. 3D representations are alsovaluable in discussing the absorption of electromagnetic radiation as a function of thealignment of transmitter and receiver.
The radiation pattern of a vertical dipole is equal in all directions perpendicular to theaxis of the antenna. In ﬁgure 4 two frames are shown, also indicating the magnetic ﬁeld by Hlines in the equatorial plane.
The animated visuals illustrate that the magnetic ﬁeld lines are concentric circles andalways perpendicular to the electric ﬁeld. Both of these ﬁelds are perpendicular to thedirection in which the wave is travelling. The ﬁelds vary with the same frequency and inphase with each other in the far ﬁeld, which can be seen when the display window is enlarged
to 2λ or 4λ. The far ﬁeld pattern shows a transverse electromagnetic wave, with electric andmagnetic ﬁelds at right angles to each other and at right angles to the direction of propagation.
The electric polarization is coplanar with the direction of the dipole vector.
3.3. Detachment of field lines, reversal of the dipole and zero-crossing of the electric field

To discuss the separation of ﬁeld lines from the dipole, the strength of the electric ﬁeld in theequatorial plane is plotted on top of the ﬁeld lines in ﬁgure 5 as a line graph. In thisillustration it is apparent that the ﬁeld lines detach from the dipole when it reverses itself andthus the ﬁeld strength directly beside the dipole has a zero-crossing.
This becomes more familiar by considering a simple dipole built up by two equal chargesof opposite sign. These charges deﬁne the start and end points for electric ﬁeld lines, and they6

Eur. J. Phys. 37 (2016) 065206

R V Girwidz

Figure 5. Drawing of ﬁeld lines combined with a diagram to indicate the value of E in
the equatorial plane.

Figure 6. 3D plot of ﬁeld lines, combined with a diagram to indicate the behavior of E

and H in the equatorial plane.

oscillate up and down in alternating directions harmonically. As they approach one another at
the midpoint of the system, the start and end points of the ﬁeld lines coalesce. As the charges
cross the midpoint, they neutralize one another, leaving no net charge to terminate electricﬁeld lines. Thus they form closed loops. Thereafter, the direction of the dipole vector as wellas the direction of circulation of the ﬁeld lines is reversed.
3.4. Electric and magnetic field—phase shift in the near field

The electric and magnetic ﬁelds near the oscillating dipole/antenna are quite different fromthe ﬁelds at greater distances. Close to the dipole, there is a phase shift between the alternating
electric and magnetic ﬁeld. When the electric ﬁeld reaches its maximum the magnetic ﬁeld
becomes zero, and vice versa. Conversely, far away from the antenna, the electric andmagnetic ﬁelds oscillate in phase with one another, perpendicular to each other and to the
propagation of the electromagnetic wave. The strength of the electric and magnetic ﬁelds on
the horizontal x-axis are indicated by line graphs (see ﬁgure 6). Thus, it can be seen that close
7

Eur. J. Phys. 37 (2016) 065206

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Figure 7. Poynting vectors at two different times.

to the dipole there is a phase shift of almost π/2 between the electric and the magnetic ﬁeld.
With increasing distance, the phase shift decreases more and more. In the far ﬁeld theoscillations are in phase.
Why is the phase shift changing? To understand this, the following considerations andillustrations on the ﬂow of energy will be useful.
3.5. Poynting vector and flow of energy

According to Poynting the directed energy ﬂux density can be calculated by the cross productof E and H, using equation (6):S =E´H(6)
Figure 7 presents snapshots from an animation, displaying Poynting vectors in various
positions. The thickness and colors indicate the magnitude of the Poynting vectors.
It should be clear that the directions of the vectors vary strongly with time especially inthe near-ﬁeld. These vectors are even transiently directed towards the dipole; this is due to thephase difference between E and H. For the horizontal plane the situation is quite simple:
During periods when E and H have different signs, the Poynting vector has a negative sign,and energy ﬂows back toward the dipole. This happens, for example, when the dipole vector
is just building up again after a zero crossing. When the dipole vector is zero, the electric
current is at its maximum, as is, therefore, the surrounding magnetic ﬁeld. However,according to Lenz’s law, the magnetic ﬁeld and the electric current are linked. The electric
current cannot collapse instantly when the oscillating electric charges pass their midpoint andthe electric dipole disappears. The electric current is sustained and the energy for that ispartially recovered from the surrounding ﬁeld. Hence, the Poynting vectors brieﬂy indicate a
ﬂow of energy back to the dipole. Due to the ﬁnite velocity of signal propagation, this isrestricted to the near-ﬁeld. There is no energy ﬂow back towards the dipole in the far ﬁeld andthus also the phase shift between E and H dissolves with increasing distance.
To visualize this, a representation is available which focuses on the situation in theequatorial plane (perpendicular to the dipole). The energy current densities are shown byarrows, and the values of E, H and S are plotted in line graphs as functions of time anddistance from the dipole (see ﬁgure 8). Very near to the origin, the E and H ﬁelds are rather
strong but almost 90° out of phase. The cross product of the electric and the magnetic ﬁeld
vectors indicates the direction in which the wave travels. Thus, the phase shift has a striking
impact on the ﬂow of energy. At certain times, when the sign of the Poynting vector is8

Eur. J. Phys. 37 (2016) 065206

R V Girwidz

Figure 8. Magnitude of the Poynting vector S shown in a line graph over distance(black) together with the magnitudes of E (red) and H (light blue) in the equatorial
plane. Furthermore, the ﬂow of energy is indicated by arrows.

negative, energy is returned to the oscillating dipole. The animation is meant to make theinterplay of several physical quantities clear.
It should be observed that more than one wavelength away from the dipole E and H
oscillate almost in phase. Hence, a signiﬁcant reverse ﬂow of energy can be observed only inthe near ﬁeld. This can be seen even more clearly by displaying the Poynting vectors for alarger area in the program.
3.6. Half-wave dipole and electric-dipole radio antenna

The simplest and most common antenna is composed of two straight rods or wires aligned on
the same axis. In this example, a sinusoidal alternating voltage is applied to the wires, causing
electric charges in the antenna to oscillate. The antenna serves as a resonator, with standing
waves of electrical current ﬂowing back and forth between its ends. Thus, electric dipoles
emerge and disappear periodically. The length of the wire or rod determines the wavelengthof the electromagnetic radiation that is emitted. The whole antenna is one-half the wavelengthof the radiation.
For the current, a standing wave is assumed, approximately sinusoidal along the length ofthe dipole, with a node at each end and maximum amplitude in the middle. Such a linear
antenna may be treated as a series of very short conductors. Based on the superposition
principle, a mathematical expression for the radiation of a linear antenna can be derived fromthe ﬁeld equations of a Hertzian dipole (e.g. see [7, 8]).
At close ranges, there are signiﬁcant differences between the wire example considered
above and the Hertzian dipole. In the far ﬁeld, the differences between the Hertzian dipole andthe half-wave dipole are smaller. This can be shown using the dipole simulation program (seeﬁgure 9).
3.7. Electric field lines as contour lines of a potential function

Differential calculus makes it possible to derive a potential function, with the electric ﬁeld
lines being contour lines of this potential ﬁeld [9]. The values of this potential function f can
9Eur. J. Phys. 37 (2016) 065206

R V Girwidz

Figure 9. Snapshots from an animation showing momentarily ﬁelds of a radiating half-wave dipole.

10Eur. J. Phys. 37 (2016) 065206

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Figure 10. Snapshots showing the potential function and the corresponding lines of theelectric ﬁeld.

be computed using equation (7)f=⎡ cos (wt - k 0 r + j0 )⎤E0sin2 J ⎢- sin (wt - k 0 r + j0) ⎥ .
⎣
⎦
k0
k0 r

(7 )

See [9, p 226], where r and J describe the position in polar coordinates, w and k 0 are theangular frequency and angular wave number, and E0 and j0 are constants to represent theinitial conditions.
In ﬁgure 10, snapshots from a corresponding animation are shown. This illustration canalso offer a meaningful context to clarify when and why the ﬁeld lines detach from the dipole
and form closed loops.
Though these three-dimensional images can fascinate physicists and even laymen, caution is advised in order to avoid misinterpretation and misconceptions. The above quantities
are computed only in the equatorial plane, with the values of the potential function shown asdisplacements perpendicular to this plane. This should not be confused with the behavior of a
two-dimensional surface wave (such as, for example, a water wave). In this context, limits ofspatial imagination should be considered.
11Eur. J. Phys. 37 (2016) 065206

R V Girwidz

4. Conclusions
The different animations above can be tools to vividly illustrate various aspects of electromagnetic dipole radiation. However, the animations should be used to highlight only the one
or two speciﬁc aspects that they are designed to illustrate. Beyond this, these visualizations
should be used with caution. They are theoretical constructs, used to illustrate certain properties and processes that are not visible in principle. Critical thinking is always required.
4.1. Field line drawings and their dynamic presentations

In general, it should be noted that ﬁeld lines and ﬁeld line images are theoretical constructs.
They help to describe spatial distributions of physical quantities. However, assigning genuine
mechanical properties to them is dangerous. This can be seen, for example, in the case ofelectromagnetic wave propagation and its representation in the form of ‘travelling’ ﬁeld lines
with speciﬁc velocities assigned to line sections.
It makes sense, in the far ﬁeld, to take the zero crossings of the ﬁeld strengths in theequatorial plane as points of reference that move at the speed of light. This can lead to ameaningful notion of signal propagation in the outer region (in the far ﬁeld). However, in thenear ﬁeld, the waves are extremely inharmonic, and it would be wrong to associate a conventional speed to a ﬁeld line segment. A rubber band, for example, would be a misleading
analogy for ﬁeld lines.
Nevertheless, there are strict limits (e.g. the dotted line in ﬁgure 2) on the areas of ﬁeld
lines with the same sense of circulation. For the Hertzian dipole, these are half spherical shellsthat are relatively easy to calculate. Their radius grows with the speed of light.
To summarize: modern media can help to illustrate many characteristics of dipole
radiation. However, when using a tool such as the one described above, one should always
take into account its limitations.
4.2. Computing dipole radiation

The program described above is a HTML5 application and runs on computers, tablets andsmartphones. Ready-made animations (animated gifs) are also available to overcome a limited
computing power (which is especially useful for the case of smartphones).
For more details and downloads see: http://www.en.didaktik.physik.uni-muenchen.de/multimedia/dipolstrahlung/index.html.
References[1] Jackson J D 2001 Classical Electrodynamics (New York: Wiley)[2] Grifﬁths D J 2013 Introduction to Electrodynamics (Boston, MA: Pearson)[3] Greiner W 1998 Classical Electrodynamics (New York: Springer) p 432[4] Tipler P A and Mosca G 2008 Physics for Scientists and Engineers (New York: W H Freeman andCompany)[5] Hecht E 2000 Physics: Calculus (Paciﬁc Grove, CA: Brooks/Cole)[6] Ainsworth S E 1999 A functional taxonomy of multiple representations Comput. Educ. 33 131–52[7] Schäfer K and Kuhn W 1986 Strahlungsfeld und Richtcharakteristik von Antennen Praxis derNaturwissenschaften. Physik 35 (6) 13–21[8] Zuhrt H 1953 Elektromagnetische Strahlungsfelder (Berlin: Springer)[9] Kark K W 2014 Antennen und Strahlungsfelder (Wiesbaden: Springer)[10] Wikipedia pages: https://en.m.wikipedia.org/wiki/Dipole_antenna (09-Sep-2016), https://de.
wikipedia.org/wiki/Hertzscher_Dipol (09-Sep-2016)12Eur. J. Phys. 37 (2016) 065206

R V Girwidz[11] MIT web pages: http://web.mit.edu/viz/EM/visualizations/light/DipoleRadiation/DipoleRadia
tion.htm (09-Sep-2016)[12] Radiation from half wave antenna on YouTube: https://m.youtube.com/watch?v=Wllxhuo9iPk(09-Sep-2016)[13] Electric dipole propagation pattern and means to concentrate the radiated energy: https://m.
youtube.com/watch?v=UOVwjKi4B6Y (09-Sep-2016), https://youtube.com/watch?v=md7
GjQQ2YA0 (09-Sep-2016)13