Journal of Physics Special Topics
P3 7 E.M. Emitter
K. Tassenberg, B. S. Chima, B. M. Lloyd, N. Wall
Department of Physics and Astronomy, University of Leicester. Leicester, LE1 7RH.
November 12, 2015.
Abstract
In the film ’Batman Begins’, a ’microwave emitter’ is used to vaporize water. The absorption coefficients
of microwaves and short wavelength infrared (SWI) electromagnetic waves in water are compared and
the absorption coefficient of microwaves in water is found to be about 200 times less than the absorption
coefficient of SWI in water, as such SWI photons are far more efficient in heating up water.
is equal to 7.748×10−20 J per molecule, or 0.484 eV per
molecule. Using this value of energy, the electromagnetic frequency of a photon that can bring the water
to boiling point in a single absorption can be found by,
Introduction
From the film Batman Begins, a so called ’microwave
emitter’ is used to turn the liquid water to vapour.
This fictional microwave emitter is strong enough to
vaporize almost all water within approximately a radius half a kilometre in a short period of time and only
vaporizes water sources, such as within water lines or
drainage pipes and does not affect people. Note that
exposing skin to the frequency of 2.45GHz [1], that of
a microwave oven, can cause serious injuries; an exposure of 5 seconds of a hand in a microwave oven was
enough to cause second-degree burns [2].
The effect of microwaves and other frequency waves
will be investigated.
f=
I = Io e−αS ,
NM
c∆T,
Av
Discussion and Results
With the equations set up, the minimum intensity required can be calculated. Assuming a cube of water
with each side 10 cm, the mass of the cube would be
one kilogram. One kilogram of pure water equates to
3.34 × 1025 molecules, which is also how many photons
at 0.117 PHz are required to turn the cube of water
into steam. Assuming that the photons have a square
flux in the direction normal to a face of the cube with
an exact overlap, this would mean that there would
have to be 3.34 × 1028 photons per m2 .
Compared to microwaves at 2.45GHz, the photons
required to vaporize 1kg of water, for which 2.59 × 106
J (using c∆T + Lv ) is required, is 1.595 × 1032 photons
per m2 . This is 4775 times more than the SWI photons.
Therefore for microwaves to be comparable, the SWI
intensity absorbed by water has to be 48 times less
than the microwave intensity absorbed by the water.
As such, for the cube of water the absorption coefficient
has to be
(1)
(2)
where c is the specific heat capacity of water, 4180
J kg−1 K−1 [3, pp. 593], and ∆T is the change in
temperature which in this case is 80o C.
This means that the energy per molecule,
dQt
M
=
(c∆T + Lv ),
dN
Av
(5)
where α is the absorption coefficient, I is the intensity
after a distance S through the medium and Io is the
original intensity. The assumption in eq. (5) is that α
is constant throughout the medium, which is a reasonable assumption in this case as the water is taken to
be clear throughout.
where m is the mass of water, N is the number of water
molecules and M is the molar mass of water, 0.018kg
mol−1 [4], and Av is Avogadro’s constant, 6.02 × 1023
molecules mol−1 .
The other equation required is the one utilizing specific heat,
Q = mc∆T ∴ Q =
(4)
where h is the Planck constant and f is the frequency.
Thus the frequency of the photon was calculated to
be 0.117 PHz, which is within the short-wavelength
infrared (SWI) range.
The second order of operation is to examine the intensity function [5],
Theory
First, the energy required for water to boil from room
temperature and pressure will need to be calculated.
This means taking into account the latent heat of vaporization of water, Lv , which is a constant value of
2257 kJ K−1 [3, pp. 596] and the temperature change
from RTP, 20o C to 100o C. As such,
NM
Qv = mLv ∴ Qv =
Lv ,
Av
QT
,
h
(3)
1
E.M. Emitter, November 12, 2015.
αM
Io−m
),
= αI + 10log(
Io−i
data on the refractive index for different wavelengths
could not be obtained.
The maximum absorption coefficient in this data occurs at 69nm/4.34PHz/18.01eV [7], which is in the extreme ultraviolet range and is entirely ionizing radiation. The standard enthalpy of formation of water is
-2.97eV per molecule [8] and as such, the ultraviolet
radiation will be able to break the water molecule itself apart. Not to mention that the ionization energy
of hydrogen is 13.6eV [3, pp. 1233].
(6)
where Io−m is the photon intensity of the microwaves,
Io−i is the photon intensity of the SWI, αM is the absorption coefficient of the microwaves and αI is the abis therefore 4775,
sorption coefficient of the SWI. IIo−m
o−i
so αM has be 8.47 more than αI for the microwaves
to be comparable to SWI. Note that this assumes that
the intensities calculated from eq. (5) are the same for
both, so IM and II are equal.
To determine which is frequency is more effective,
the next thing to do is to find the real absorption coefficients for these wavelengths in water.
Conclusion
It was known from the beginning that the ’microwave’
emitter would seriously damage someone, as approximately 70% of the human body is made of water. As
such, the only reason to use the emitter is when damaging people is not a problem; So the most efficient way
to heat up water is using short-wavelength infrared.
There is much more left to discuss about such a
device, for example, the sort of intensity required to
achieve the, approximate, half a mile range or finding the absorption coefficients of other materials and
determining the effect of these.
At the very least, we have determined that Lucius
Fox (a character in the movie Batman Begins) does
not know what a microwave emitter is and must have
designed something completely different.
References
Fig. 1: This graph is a plot of the results from an article
written by D. J. Segelstein, obtained from ref. [6]. This
graph is a plot of ref. [7].
[1] T. Koryu Ishii. Handbook of Microwave Technology,
page 287. 1995.
[2] L. A. Geddes and R. A. Roeder. Handbook of Electrical
Hazards, page 370. 2005.
[3] Paul Allen Tipler and Gene Mosca. Physics for scientists and engineers with Modern Physics. W.H. Freeman, 2007.
[4] water — H2O - PubChem. http://pubchem.ncbi.
nlm.nih.gov/compound/water#section=Top.
Accessed: November 12, 2015.
[5] R. Siegel and J. Howell. Thermal Radiation Heat Transfer, page 425. Taylor & Francis, 4th edition, 2001.
[6] S. Prahl.
Optical Absorption of Water Compendium.
http://omlc.org/spectra/water/abs/
index.html. Accessed: November 12, 2015.
[7] D. J. Segelstein. Raw data from ’the complex refractive index of water’. http://omlc.org/spectra/
water/data/segelstein81.dat. Accessed: November
12, 2015.
[8] D. A. Johnson. Metals and Chemical Change, volume 1,
page 65. Royal Society of Chemistry, 2002.
The wavelength of the photons from the microwave
oven is 0.12 m and the wavelength of the photon that
would vaporize a water molecule with a single absorption is 2.56 µm. From figure 1, it is easily noticeable
that the absorption coefficient of the microwave oven
(α = 0.5431 [7], using a straight line fit) is far less than
the absorption of the SWI photon (α = 111.21 [7]).
A fraction of the energy of the microwave photons was able to cause second-degree burns, so whilst
switching to a higher energy photon would have better
efficiency at heating up water, it would also be so much
more dangerous to people around. By using eq. (5) for
both the microwave and SWI photons, and setting Io
and S to be the same for both, a relation,
IM
IM
= eS(αI −αM ) ∴
= e110.69S ,
II
II
(7)
can be obtained where IM is the intensity of the microwave photons and II is the intensity of the SWI
photons. It is obvious that with such a difference in α
values, eq. (7) is very sensitive; after 10 cm of going
through water, IM is 6.42 × 104 times bigger than II .
Note that a more accurate comparison can be made
if the transmission fraction is available for both, but
2