Microwave Experiment Phys2055

Christopher Raymond 42188207, Kyle Muzyczka

4/18/2011

Microwave Experiment

Abstract: This experiment is to investigate several properties of waves that can be measured on a macro scale using microwave radiation. Once these properties have been analyzed, experimentally calculated wavelengths were compared to a theoretical value of 2.86 cm. The two experimentally determined values for the wavelength were 2.92cm and 2.49cm, which are within 2% and 13% respectively, of the theoretically determined wavelength.

Introduction: The aim of this experiment was to measure various features of transmission of microwaves, and then relate interference patterns to the wavelength of the radiation.

Theory: Electromagnetic radiation is a wave that creates an electric and magnetic field as it travels. These electromagnetic waves can be reflected off surfaces, for example light. The energy of the wave can be attributed to several factors, one of them is the wavelength of the radiation. A single wavelength is the distance from one peak to the next consecutive peak of the wave.

For this experiment, microwave radiation is being used. This allows the apparatus to be scaled up in size, from what might normally be used when analyzing light, which has a much shorter wavelength. By using the equation � � �� �, where c is the speed of light, f is the frequency of the wave and λ is the wavelength, it is possible to get a theoretical value for the wavelength of the radiation emitted by the transmitter in the experiment. The frequency quoted on the transmitter was 10.5 GHz, using that value, and c = 3x10^8 m/s, the theoretical value for the wavelength of the radiation emitted was 2.86cm.

The radiation used in the experiment bounces off objects easily at short distances. This meant that there was a standing wave interference pattern formed in the area between the transmitter and receiver. Standing wave interference occurs when an incoming wave, output from the transmitter, interacts with reflected radiation, coming back in the opposite direction, after bouncing off the receiver. This interaction between the two waves is characterized by the two electric fields created by the waves interacting. There are points of constructive and destructive interference in standing waves. Points of constructive interference occur when the electric fields have components that add in the same direction. This causes an area where the electric fields in that space (and waves propagating through it) have greater intensity. Destructive interference occurs when the opposite interaction takes place between the transmitted and reflected waves. When the electric field components from the two waves are in opposing directions they cancel each other, reducing the strength of the overall field, and the intensity of the wave. These effects are noticeable in this experiment, and occur in areas referred to as nodes (constructive) and anti-nodes (destructive).

The double slit section of the experiment allows measuring of the nodes and anti-nodes created by diffraction patterns. Diffraction occurs when a wave is passed through a gap where the width of the gap is less than the wavelength of the wave. The wave appears to begin semi-circular propagation after

passing through the slit. When two slits are placed next to each other, the waves propagated by both slits begin to interact, in the same way that occurs in the standing wave case. The nodes appear at angles according to the equation:

����� � �

Where d is the distance between the slits.

Lloyd’s mirror causes standing wave interference by reflecting radiation off a surface parallel to the direction of transmittance. By varying the distance of the reflecting plate, the field of nodes and antinodes changes.

Method: The microwave experiment involved firing microwave radiation from a transmitter, to a receiver. The experiment was conducted in three main parts;

The first part was to measure the effects of moving the receiver on the intensity of the incoming wave. The first measurements were taken in order to measure the effect of moving the receiver directly away from the transmitter. An effect was noticed during this part of the experiment, which showed there was a standing wave interference pattern in the data. Because of this, data was taken at both very short, and long intervals. The very short intervals (2mm) are taken to try to show that there is a standing wave interference pattern between the transmitter and the receiver, and the longer intervals (5cm) are to plot the decay of intensity with respect to distance.

Effects due to angle were also measured, so to plot the fall of intensity as the angle of the receiver is changed. This can also give an idea as to the emitted radiation field of the transmitting antenna. Radiation intensity was measured while rotating both the receiver about its base, and rotating it around the goniometer.

Part 2 of the experiment involved investigating double slit interference effects. This was done by creating a double slit using a series of magnetized plates. The distance between the slit was varied for 3 different runs, and the slit spacing was kept constant at 1.5cm for each trial. By tuning the receiver to a maximum at the 180 degree point of the goniometer, the interference pattern could be measured by rotating the receiver around the center-point on the apparatus where the slits had been placed. There is a point that holds the slits in place on the goniometer.

Part 3 of the experiment investigated Lloyd’s mirror, and the effects of placing a reflector parallel to the axis between the transmitter and receiver. Measurements of how far away the reflector was from the axis, were taken at two distances where the receiver registered a maximum point. These points can be compared in order to give a wavelength for the radiation from the transmitter.

Analysis: The graphs used in this section were generated using Matlab, with my experiment partner Kyle.

The initial graph shows the effect of distance on the intensity of the radiation. Note that the data displayed on the graphs has been normalized. This has been done so that there is a constant measure of intensity, that is uniformly applied over all the data recorded, when it comes to graphing. The first graphs of how much the intensity falls off with distance, were measured between 55-90cm at 5cm intervals with an error 0.2 cm. Intensity could be measured to within 0.1, on the meter attached to the receiver. These plots use data found in table 2, contained in the appendix.

For the above graph the distance data was also normalized. As you can see the error in this portion of the experiment was very low, as the data could be read with high precision. The graph is showing that the intensity falls off at 1/R^2, and showing point source style radiation emitted from the transmitter.

By fitting these points with a quadratic line the following graphs are generated.

The initial plot shows the relationship of the points to a standard x^2 polynomial fit. The second graph shows the points, plotted along-side of a 1/R^2 graph. This shows that the actual data fell off slightly faster than 1/R^2 in the experiment. This falling off faster could be because of the reflected effects and the standing wave.

The next graphs show the standing wave patterns formed in the area between the transmitter and receiver, due to reflected radiation.

Plots use data from table 3. These two plots are of the data taken from 57cm to 60cm taking measurements every 2cm. The first plot shows the error on the data, and the second shows a Sine curve plotted as a fit by Matlab. It is very clear that there is a sinusoidal standing wave formed over small distances. It’s clearer in the second graph that there is still an overall drop in intensity from the beginning to the end of the measured area. This can be interpreted as the intensity of the wave dropping off with distance. The final peak of the function in the plot is substantially lower than the first, and this effect is the same for the first and second minimum points.

Plots use data from table 1. These next two plots show the same effect using the data taken 87-90cm away from the transmitter. The second fit line plot shows even more clearly the gradual decline of the sine wave due to the extra distance. This means that the wave travelling in the experiment can be modeled as decaying as

�������

Where alpha is a modifier that relates the frequency of the sine wave to the wavelength of the sine wave (in this case the wavelength was approximately 1.4cm). Note that the sine wave still decays as 1/R^2.

The effect of rotating the receiver on its own point shows the drop off of radiation intensity due to its polarization. Plot uses data from table 4.

From this graph it’s very easy to see that as the rotation of the receiver approaches π/2, there is no more intensity reaching it. This effect is well within expected values, and shows the polarization of the waves very well. The point 0 radians is the upright position, then the receiver was slowly turned through π/2 radians to the point where there was no longer a visible reading on the receiver.

The remaining section from part A of the experiment involved moving the receiver about the rotation of the goniometer. This experiment allowed very interesting results to be seen about the shape of the emitted field from the horn antenna. The first two plots show the radiation intensity against radians, where the first has error values taken from the measurements, and the second has a fit line generated in Matlab. The second two plots are polar graphs also generated using Matlab that clearly show the field of the antenna. This was done by plotting intensity as the radial length and angle in radians onto the plot. All plots use data from table 5.

It’s very easily shown that the intensity gradually falls off as the angle is increased to approximately 0.4π radians either side of the axis of transmission. The next two plots take this data and show the polar relationship between the data.

The raw data can be seen here as a narrow band within a 60 degree arc. It’s very clear that the intensity drops to 0 as the angle approaches π/6 radians either side of the zero. Using this data we can see the field created by the antenna.

This graph is in accordance with the standard emission pattern from a single horned antenna. The data plotted well in this case and clearly shows the relationship between angle and intensity. One interesting point to note is that there is slightly higher intensity between 0-330 on the polar plot, when compared to the other side. This is most likely due to the effect of the wall next to the apparatus causing some of the radiation to be reflected back. This would also explain why this effect occurs on this side of the graph, because between 0-30 degrees, the receiver is pointing away from the wall where there might be some reflection of the radiation.

DOUBLE SLIT

The double slit portion of the experiment caused some very interesting results. The experiment was completed three times, each with different distances between the slits. The first time the experiment was run, the results showed defined individual peaks, this experiment used slit spacing of 5.6cm (error of 0.1cm)

Plot uses data from table 6. One point to notice is that the peak on the right hand side is comparable in intensity to the main peak of the pattern. This is most likely because of reflected radiation off the wall which is interfering with the results on the right hand side of the goniometer. The other two runs of the experiment yielded unusable results. The results from these runs are tabulated below.

part 1 Angle (degrees)

Error (degrees)

Intensity (normalized)

error

125-135 5 0.1 0.02 145-155 5 0.1 0.02 160-170 5 0.1 0.02 190-200 5 0.1 0.02 205-215 5 0.1 0.02 225-235 5 0.1 0.02 part 2 130-145 5 0.1 0.02 155-165 5 0.1 0.02 195-205 5 0.1 0.02 215-230 5 0.1 0.02

Part 1 refers to data taken when the distance between the slits was 10cm, and part 2 refers to spacing of 7 cm. Both of these trials caused extremely low values for intensity to be found over a range of angles. The main peak was nearly identical to that measured in the initial trial, where by the main intensity was 1.0 on a normalized scale, and fell off to 0 over a range of 5 degrees in either direction. But for all the other measurements taken at these slit distances, we were unable to get anything more than a slight rise in intensity, even on the highest setting of the receiver. There was not enough fluctuation in the intensity to get a reading, only a constant, very low, intensity over a broad range of angles.

This may be due to further interference effects, but is most likely due the loss of contrast evident in the double slit experiment as the distance between the two slits is widened.

In order to calculate the wavelength from the double slit part of the experiment, the equation:

����� � �

was used, along with the small angle approximation. The small angle approximation states that sin(x) = x, and this is a valid approximation in this case (changing this makes the double slit appear as if its travelled onto a flat surface at the distance of the receiver which doesn’t appear in the formula above), and allows the wavelength to be expressed as:

�� � �

Using the second peak with the small angle formula, λ = 2.92+/- 0.3cm, which when compared to the expected value, generated theoretically from the frequency of the transmitter, is out by approximately 2%.

For the Lloyd’s Mirror section of the experiment, data was taken over two positions:

Lloyd's Mirror Distance (cm) error h1 17.3 0.2 h2 20.8 0.2

For this data, the distance between the receiver and the transmitter was 100+/- 0.2 cm. Using the equation:

�� � �� � �

Where s is the distance that the reflected waves have to travel, x is the distance from the transmitter to the axis of the mirror, and n is an integer attached to the wavelength. We can see that major nodes appear at the points where the difference between the reflected distances are multiples of wavelengths. So this means that:

�� � �� � �

Where

���� � ��� � ���� So by this trigonometric relation, we arrive at a value for λ of 2.49+/-0.02cm. This value for the wavelength differs from the theoretical value by 13%.

Discussion:

This experiment had some interesting results, especially in the first part of the results, looking at how the intensity falls off with distance. The standing interference wave that was found in the first part of the experiment had a wavelength approximately half of the transmitted wave used in the experiment. This is most likely due to the reflection of the wave of nearby surfaces. It was easy to see that the experiment was sensitive to other sources of reflection. This can be seen in the results for rotation of the goniometer and the double slit results, wherein the data recorded when the receiver was facing the nearby wall, had slightly stronger intensity then that measured facing away from the wall. It was also evidenced, although not in the results, by large fluctuations (normalized to between 0.2 and 0.4 points on the intensity scale of 0-1) when a person passed within a reasonable distance (approximately 50cm) of the back of the receiver. The reflected waves could have created a separate series of nodes and anti-nodes. Another possible explanation is that the wave reflected off the receiver is slightly out of phase. Because the measurements of the standing wave interference only being conducted over such a short interval (3cm), there wasn’t enough data to conclusively say whether or not a phase shift in the reflected wave was enough to cause the change. Another point in evidence of the phase shift idea is that the fluctuation in amplitude of the sine wave is quite small, but this could be due to loss of energy following the reflection of the incoming wave.

The double slit experiment had an interesting problem arise with it. During the trials when the slit spacer was widened (7cm and 10cm), there was no usable data generated in the experiment. The intensity was too low to be able to get data for the fluctuation of intensity over an angle, but there was enough a change in intensity to see that there was at least something going on over a large range of degrees. This loss of intensity is most probably due to the loss of contrast when the spacing between slits is increased. It can be seen that as the distance between the slits is increased, there are more areas that register an intensity, but the areas seem to become more difficult to define readings for. It might become easier to get usable data in the case of optical light, where the interference pattern can be seen, but in this case it was too difficult to find exactly where the nodes formed, and this meant that there was no way to deduce a wavelength for these trials.

The Lloyds mirror experiment was very smooth, there were no large errors that seemed to be evident in the results. The difference in the wavelength from this experimental value to the theoretical, could have maybe been caused by using too large a reflecting plate. With a plate that is too wide, there would be overlap between the incoming waves, also there may have been more interference with other reflected waves.

Ideally the entire experiment should be done in an area that is microwave absorbent, so to ensure that there is no reflection from nearby walls and objects. This would eliminate much of the experimental error, especially in the case of the goniometer rotation and the double slit experiment.

The actual values achieved experimentally for the wavelength of the radiation was very close to the theoretical value. In the case of the double slit experiment the difference between the two values was only 2%. This number is very accurate, and a surprising result, given that there was a large amount of fluctuation in the intensity of the double slit experiment. In the Lloyd’s mirror section, the error of 13% seemed quite reasonable. Given the uncertainties, and the amount of interference occurring in the experiment, the value of 13% seems allowable, and confirms the validity of the Lloyd’s mirror trials.

In conclusion the experiment was definitely successful. The initial part of the experiment showed quite well that not only does the intensity pattern drop of at a rate of 1/R^2, but there is also a standing wave interference pattern that forms between the transmitter and the receiver. The effects of polarization were easily seen in the data of when the receiver was rotated around its own base, and the emitted field of a horned antenna was measured during the rotation of the receiver about the goniometer.

The Double slit and Lloyd’s mirror sections of the experiment confirmed the wavelength of the radiation to a substantial level of accuracy and in the case of Lloyd’s mirror, a good level of precision (error was only about 1% in the final reading).

APPENDIX:

All raw data can be found in the appendix:

distance (cm) error (cm)

intensity error

90 0.2 2.95 0.1 89.8 3.05 89.6 2.7 89.4 2.3 89.2 2.1 89 2.25 88.8 2.65 88.6 3.2 88.4 3.35 88.2 3 88 2.6 87.8 2.3 87.6 2.4 87.4 2.8 87.2 3.3 87 3.55 Table 1: is for the data taken at short intervals, long distance from the transmitter.

distance (cm) error (cm)

intenstiy error

90 0.2 2.95 0.1 85 2.8 80 4.25 75 4.2 70 5.25 65 6.8 60 7.6 55 10 Table 2: data taken at long intervals.

Distance (cm) error (cm)

intensity error

60 0.2 7.5 0.1 59.8 8.5 59.6 9 59.4 8.8 59.2 8.3 59 7.9 58.8 7.6 58.6 7.95 58.4 8.45 58.2 9.6 58 9.5 57.8 8.8 57.6 8.4 57.4 8 57.2 7.95 57 8.8 Table 3: data taken at short intervals, small distance from transmitter.

angle (degrees)

error intensity error

0 1 8.9 0.1 5 8.75 10 8.6 15 8.35 20 7.95 25 7.45 30 6.75 35 6 40 5.5 45 4.05 50 3.1 55 2.05 60 1.1 65 0.5 70 0.2 Table 4: Data for the rotation of the receiver about its base.

angle (degrees)

error intensity error

160 0.5 0.4 0.1 162 1.4 164 2.35 166 3.65 168 4.5 170 5.35 172 6.4 174 7.45 176 8.1 178 8.55 180 8.8 182 8.6 184 8.05 186 7.05 188 5.9 190 4.9 192 4.05 194 3.05 196 1.95 198 0.95 200 0.4 Table 5: rotation of receiver about the goniometer.

angle (degrees) error (degrees)

intensity error

145 0.5 2.4 0.1 150 0.5 4 0.1 155 0.5 2.2 0.1 165 0.5 0 0.1 170 0.5 0 0.1 175 0.5 3.8 0.1 180 0.5 5.5 0.1 185 0.5 4 0.1 190 0.5 0 0.1 205 0.5 1.4 0.1 210 0.5 5.6 0.1 215 0.5 2.6 0.1 220 0.5 0.6 0.1 225 0.5 0 0.1 Table 6: data for intensity of the double slit experiment.

EQUATIONS:

���������� ! � "�#���$�� !��������� ! � %& '(( � #�����%���������

Error equations used:

)# � *)��� � )+��

)#,#, � -.)�� /� � .)++ /�

)# � % ,����,)�