Propagation of microwaves
(inverse square law)
TEP
Related topics
Microwaves, electromagnetic waves, spherical waves, virtual source, reflection
Principle
The intensity of a source of radiation, e.g. a microwave transmitter, at any given location
depends on the distance of this location from the (approximately punctiform) source.
Actually, due to its antenna geometry, a microwave transmitter can only be regarded as a
punctiform source of radiation at long distances, which is why it is described as a virtual
source for shorter distances.
Note
Prior to performing this experiment, it would be helpful, though not mandatory, to perform
the experiment P2460310 "Reflection, transmission, and refraction of microwaves".
Equipment
1
1
1
1
1
1
1
1
1
1
Tasks
Microwave set 11742-93
Microwave transmitter
Microwave probe
Microwave control unit
Meter rule
Additional equipment
Multi-range meter, analogue
Connecting cord, 32 A, 750 mm, red
Connecting cord, 32 A, 750 mm, blue
Barrel base PHYWE
Support rod, stainless steel 18/8, l = 250 mm, d = 10 mm
Right angle clamp PHYWE
07028-01
07362-01
07362-04
02006-55
02031-00
02040-55
Fig. 1: Experiment set-up
Measure the radiation intensity of the microwave transmitter for various distances r and
determine the position a of the virtual source.
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1
TEP
Propagation of microwaves
(inverse square law)
Theory
The waves that are emitted by microwave transmitters are spherical waves whose wave
fronts can be described as the surfaces of spheres. During the propagation of these wave
fronts, the law of conservation of energy applies: When one of the shells propagates, the
next higher shell includes the same (total) energy with a lower energy density. Based on
the spherical geometry (radius r, surface of sphere O = 4πr2) the decrease in intensity I
results as follows:
I (r )∼
1
r2
(1)
This is the inverse square law. It applies to all spherical waves.
Since the intensity is the square of the amplitude of the electric field strength E, it
decreases as a function of the inverse distance:
E (r )∼
1
r
(2)
However, this is only approximately true for the present experiment, since the funnelshaped geometry of the source affects the propagation of the waves in the case of short
distances (reflection inside the funnel). This is why it is useful to calculate with an effective
geometry, which assumes the existence of a virtual source at the distance a from the
transmitter (see Fig. 2).
Fig. 2: Transmitter and virtual source
Accordingly, the length r must be corrected based on the position of the virtual source
when measuring the intensity.
In the case of the present set-up, the intensity is measured by way of a probe that emits a
voltage signal U that is proportional to the intensity:
1
2
(r −a)
(3)
1
∼r −a
√U
(4)
U∼
As a result, the following applies:
This relationship enables not only the verification of the inverse square law, but also the
determination of the location of the virtual source.
2
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P2460401
Propagation of microwaves
(inverse square law)
TEP
Set-up and procedure
Set the experiment up as shown in Fig. 3.
Fig. 3: Experiment set-up
Connect the microwave transmitter and probe to their associated sockets of the control
unit. Connect the multi-range meter to the voltmeter output of the control unit and select
the 10 V measuring range (direct voltage). The loudspeaker and internal or external
modulation are not required for the time being.
Mount the probe in the barrel base by way of the boss head. Ensure that the round mark
near the measuring head points upwards. Position the transmitter at the far end of the
meter rule (e.g. at 790 mm).
Fig. 4: Microwave transmitter and probe
Place the probe in the beam path close to the transmitter so that the probe is perpendicular
to the direction of propagation of the radiation and the measuring head is located directly
above the meter rule (see Fig. 4 and 5). Switch the transmitter on by connecting the
control unit to the mains power supply and set the amplitude controller to half maximum.
Check the height of the probe in its holder by varying the height of the boss head in order
to maximise the voltmeter reading. If necessary, reduce the amplitude if the selected
measuring range is exceeded when changing the position by a few centimetres along the
meter rule.
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Propagation of microwaves
(inverse square law)
TEP
Fig. 5: Reading the meter rule (for example the position r = 440 mm)
Then, measure the radiation intensity for various positions r of the probe. To do so, extend
the distance between the probe and transmitter in steps of 1 cm and note down the
reading of the voltmeter. When reading the position, ensure to look at the meter rule
perpendicularly from above without any parallax. Ensure also that the probe is always
aligned perpendicularly with regard to the meter rule, i.e. that it is not offset (see Fig. 5).
Record a total of 40 measurement values. In order to expedite the experiment, we
recommend using a larger step width, e.g. 2 cm, for longer distances. Since the intensity
changes most significantly in the case of shorter distances, it would be useful to maintain a
step width of 1 cm in this case.
As a final step, switch the internal loudspeaker of the control unit on and set the modulator
to "internal". Then, move the probe along meter rule over the entire distance and listen
closely to the volume of the signal. Note down your observation.
Note
During the experiment, do not stand in the direct vicinity of the beam path when reading
the voltmeter values. The human body reflects microwaves so that the measurement result
may be invalidated. The same applies to all types of metallic objects (see also the
experiment P2460301 "Reflection, transmission, and refraction of microwaves". If several
experiments are performed simultaneously in a laboratory, ensure sufficient distance
between the experiment stations in order to avoid interference signals caused by reflected
radiation and/or scattered radiation from the other set-ups.
Evaluation
Present the receiver signal U of the probe as a function of the distance r with regard to the
transmitter. Determine the location of the virtual source (intersection with the r-axis) by
plotting 1/√U.
4
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P2460401
Propagation of microwaves
(inverse square law)
r in mm
rrelative in mm U in V
1/√U in 1/√ V
640
630
620
610
600
590
580
570
560
550
540
530
520
510
500
490
480
470
460
450
440
430
420
410
400
390
380
370
360
350
340
330
320
310
300
290
280
270
260
250
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
0.338
0.350
0.386
0.381
0.397
0.415
0.424
0.447
0.466
0.466
0.503
0.510
0.535
0.550
0.573
0.598
0.603
0.636
0.674
0.698
0.721
0.751
0.745
0.767
0.791
0.784
0.830
0.823
0.853
0.913
0.886
0.953
0.913
0.988
1.085
1.069
1.101
1.174
1.136
1.217
8.75
8.15
6.7
6.9
6.35
5.8
5.55
5
4.6
4.6
3.95
3.85
3.5
3.3
3.05
2.8
2.75
2.475
2.2
2.05
1.925
1.775
1.8
1.7
1.6
1.625
1.45
1.475
1.375
1.2
1.275
1.1
1.2
1.025
0.85
0.875
0.825
0.725
0.775
0.675
TEP
Table 1: Example data
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TEP
Propagation of microwaves
(inverse square law)
The following representations use the distance r with regard to the transmitter:
Fig. 6: Graphical representation of the example data
The linear relationship enables the determination of the location of the virtual source as the
intersection of the straight line with the r-axis. The example data in Fig. 6 lead to the
distance a = 14.5 mm, i.e. the virtual source is located inside the funnel.
Result
There is a square dependence of the radiation intensity on the distance and location of the
virtual source. The virtual source is located in front of the actual transmitter.
Please note that a periodic oscillation (periodicity 31.58 mm), which was made audible by
way of the loudspeaker, is superimposed on the inverse square law. This periodic oscillation
is due to the fact that the housing of the probe reflects the microwave radiation which, in
turn, results in a standing wave between the probe and transmitter (metallic funnel). This
phenomenon is described in detail in the experiments P2460310 "Reflection, transmission,
and refraction of microwaves" and P2460510 "Standing waves in the microwave range". It
is a systematic error source that exists regardless of the inaccuracy when reading the scale
(meter rule) and voltmeter.
6
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