Abstract
In this experiment, we worked with the famous Michelson interferometer and explored some of its many uses. The first part of the experiment focused predominantly on building the interferometer correctly, in order to observe clear interference patterns that we could take readings from. The second part saw us using these interference patterns to test the magnetostrictive effects on various metals. Using rods of these materials, we found that nickel and iron were two metals that displayed these effects. They are also ferromagnetic materials. Consequently, we were able to make a link between metals that exhibited magnetostrictive effects and ferromagnetic materials. Finally, the third part of the experiment showed us exchanging the metal rods for a cuvette of air. This allowed us to practise another use of the interferometer, and measure the refractive index of air. Our experimental value of the refractive index was found to be .
Introduction
The Michelson interferometer was invented in the late 19th century by Albert Michelson. The tool is widely used today to measure the wavelength of light, the refractive index of transparent materials and small changes in length, which we will use to measure the magnetostrictive effect of various ferromagnetic metals. Ferromagnetism is the strongest type of magnetism and is the only type that generates forces strong enough to be felt. The magnetostriction effect is when a ferromagnetic rod, like iron or nickel, is placed in a magnetic field parallel to its length, so that the rod experiences a small change in its length. The Michelson interferometer is used to magnify the rods’ interference patterns, and make these small shifts in length observable.
Another use for the Michelson interferometer is to find the refractive index of a thin plate of transparent material. Discovering the refractive index of a material can be used for various reasons, like identifying a substance, confirming its purity and measuring its concentration. In this experiment, we will be determining the refractive index of air.
We know that the refraction index of a gas, is linearly dependent on pressure . Therefore,
If there is no pressure, this means there in an absolute vacuum and the refractive index . We can calculate the change in refractive index from the data obtained using the equation
Then we have the optical path , which can be related to the refractive index of the gas we are taking measurements from. This is done using the equation
Here we have the length of the cuvette and the refractive index of the gas confined in the cuvette . If we change the pressure in the cuvette by an amount , the optical wavelength is also changed by . This is represented by
If we start at an ambient pressure , and decrease the pressure in cuvette, we can observe fringe changes from the starting point of the interference pattern up to a pressure . Going from a minimum to a minimum relates to a change in the optical path by a wavelength . Hence, amid the pressures and , we find the optical path changing by,
We must then recognise that light will travel twice through the cuvette. Thus, using equations (4) and (5), we can obtain,
Finally, combining equation (6) with equation (2) yields,
Experimental section
Figure (1) shows the experimental setup used to carry out this experiment. To begin with, we adjusted and measured the height of the laser beam to ensure it was around 130mm. We then used the mirror M1 to adjust the beam path, and direct it at a 90° to the mirror. Once we put in M1, we quickly realised the laser beam would not travel directly through the centre of the mirror, so we readjusted the laser to approximately 115mm ± 1mm for human error. The mirror M2 was then put into position, and again the beam was focused at 90°. We began with a nickel rod, and screwed mirror M3 onto the appropriate end. The rod was then inserted into the coil and fixed into place. The shaft of the coil was then placed into a magnetic base and arranged so that the mirror plane was perpendicular to the propagation direction of the laser beam. The shaft was then tightened and fixed into place. After the initial setup, we checked to ensure the laser’s beam was reflected and coincided with its point of origin from the laser. This was done by fine tuning and adjusting all the mirrors and shifting the beam back into position. Next, the beam splitter BS was placed into the setup in a way that made sure one beam still reached mirror M3 and the other was reflected at 90° from the beam splitter. A screen SC was put behind the beam splitter, and a fourth mirror M4 was introduced and placed perpendicular to the beam, so that it was reflected onto the screen. Two spots appeared on the screen and using M4 we were able to make them coincide, so we could observe one flickering red spot. Finally, the lens L was placed between the mirrors M1 and M2, and adjusted so that the beam was illuminated on the screen and we could see a clear interference pattern in the form of concentric circles. The setup took approximately 90 minutes to complete, due to having to adjust the beams to make sure they were constantly aligned every time a new element was introduced. The interferometer was also very sensitive, and we needed to ensure the bench we were working on was free of excess objects and we did not lean on or touch the bench excessively.
With the nickel rod, our first ferromagnetic material, in place, we began to test for ferromagnetic properties. This was done by setting the DC-voltage to its maximum amount and gradually increasing the DC-current in steps up to approximately 3A. The measurements were to be taken quickly, as the rod could heat up and produce extra changes in the interference pattern. We counted the changes n, from a maximum to a maximum from one point in the interference pattern. After a trial run, we realised the fringes were too small to be observed on the screen. Due to this, we decided to set the change n as the dark spot in the centre of the pattern and read off the current. The experiment would be repeated three times to improve accuracy. After nickel, we were to use our second ferromagnetic material, iron, and repeat the experiment. Iron was then replaced for a non-ferromagnetic material, copper. The experimental setup needed to be adjusted after each material, to ensure the laser beam was still centred and reflected correctly.
After measurements were taken for the metal rods, the same experimental setup was used to measure the refractive index of air. The copper rod was removed and a measurement cuvette was placed close to the mirror M3, in line with the reflected laser beam of the Michelson interferometer. The cuvette had a hand pump attached to it, which had a pressure gauge that allowed us to read off the pressure in the cuvette. Measurements were taken by gradually reducing the pressure in the cuvette using the hand pump. When doing this, the interference rings move and we can measure the number N of new fringes that appear upon the screen. This was done by counting each time a new dark spot in the centre of the interference pattern emerged. The experiment was then repeated several times to improve accuracy, and we made sure to vent the cuvette after each measurement series. These measurements were used to determine the refraction index of air.
Results and discussion
Nickel was the first ferromagnetic material we used. The circular interference fringes observed sank with the increasing current. We increased the current from 0A to approximately 3A and counted 9 new fringes in the pattern. This experiment was repeated three times and the average current for each change was taken. To calculate the error in the average current, the equation
is used. The results can be seen in figure (2). We can see that there are missing values for the current for the sixth fringe change. This was due to the interference pattern changing of its own accord, possibly to the sensitivity of the Michelson interferometer, and grew against the sinking direction is was meant to be travelling in.
Next, we swapped the nickel rod for our second ferromagnetic material, iron. Here we had to adjust the way the changes were recorded due to the limited movement once current was applied to the rod. We found that the concentric circles begin to source, before sinking again. Because of this, we counted the sourcing circles, and then counted back once they began to sink again. The experiment was repeated three times also, and using equation (8) we can find our average errors. The results for the iron rod can be found in figure (3).
We then replaced the iron for the only non-ferromagnetic material, copper. This material showed no change in the interference pattern when the current is increased. This demonstrates that the changes in interference pattern for the ferromagnetic materials is due to magnetostrictive effects and not due to other causes.
We then plotted the iron and nickel samples with their relative change in length against applied magnetic field strength. To obtain the value for relative change in length, we use the equation
where m as it is given to us, and where is found using the equation
Here, are the fringe changes we measured and mn as given to us. Similarly, the value for the magnetic field strength is determined by using another equation.
where the number of windings , the radius of a winding m and the length of the coil m are all given to us in the manuscript. the average current we measured during the experiment. The graph of these equations is seen in figure (4).
The final part of the experiment required us to measure the refractive index of air. The pressure was read off the pressure gauge every time a change in the interference pattern took place. Figure (5) shows the data we collected in this part of the experiment. The first set of results were used as a practice round as we found the cuvette to be more sensitive than expected. This set will not be included in the average pressure. The errors for the average pressures was again calculated using equation (8). We plotted the number of fringe changes against the corresponding pressures, shown in figure (6), and used this to obtain the gradient.
Using the graphing software, we found the gradient to be . We can use this to find the refractive index of air, using equation (7), where , , , and equation (1), where . When , there is an absolute vacuum, and in turn we find .
After substituting in the appropriate values and combining equations (1) and (7), we find that the refractive index of air .
Conclusion
Reference values for the refractive index of air show . Comparing this to the experimental value that we obtained, we can see a 0.24% inaccuracy. Although the experimental value was very close to the actual value for the refractive index of air, there were still ways that the experiment could have been improved.
The main errors that occurred in this experiment were due to the sensitivity of the Michelson interferometer. The interference pattern would move of its accord, making taking fringe readings very difficult. The patterns would also react to any passers-by or any slight movements. This can be detrimental as the fringe changes are crucial to proving the ferromagnetic properties of nickel and iron, and to calculating the refractive index of air. Also, when taking measurements, increasing the current can cause the coil to heat up and produce extra changes or expansions in the interference pattern.
These errors can be reduced or eliminated by being aware of environmental factors and taking extra precautions to protect the experiment from vibrations or drafts. Repeating the experiment several more times will also reduce errors, by providing a more average result. Also, reconfiguring the experimental setup to magnify the interference pattern would make observing the fringe changes more visible and therefore increase accuracy.