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The Charge/Mass (e/m) Ratio of the Electron PHYS 0212: Introduction to Laboratory Physics Fall 2012 Abstract The experiment conducted demonstrated correlation between the charge and mass of an electron and the behavior of magnetic fields. The lab was divided into four parts. The first three parts were conducted with a compass that was used to locate the magnetic field lines around a bar magnet, a solenoid and then a pair of Helmholtz coils. As a result of these trials, our observations found evidence that magnetic field lines go from a north magnetic pole to a south magnetic pole.
These results were seen in all three magnetic devices. When determining the magnitude of the electric field in the Helmholtz coils, the ? value was experimentally determined to be 8. 522? 10-4 T/A and this value differed by 9. 378% from the expected ? value of 7. 791? 10-4 T/A. Lastly, in the fourth part of the experiment, the charge to mass ratio was calculated by bending an electron beam in side an e/m tube with the Helmholtz coils and came out to be 1. 158? 1011 C/kg, with a 34. 17% error from the expected value 1. 759? 1011 C/kg. Introduction and Theory
The easiest way to describe magnetic forces is by recognizing that unlike poles attract each other while like poles repel. Kind of like the same way an electrical positive charge is attracted to an electrical negative charge. The main difference between an electrical force and a magnetic force is while positive and negative charges have the ability to be separated, it is not possible to separate opposite poles of a magnet to create what is theoretically known as a magnetic monopole. In fact, if a magnet were cut, it would create two new magnets both with a north and south pole.
This phenomenon would be observed in each piece because a north and south pole must always exist opposite each other. Electric and magnetic fields are very similar because both can create electromagnetic waves. Each wave is perpendicular to the other, and are also perpendicular to the direction they travel. Magnetic fields can be displayed by bar magnets, solenoids, and Helmholtz coils. In this lab will examine each of these objects and devices to determine how each relates to and affects a magnetic field. Part I – The Bar Magnet
As a general rule, when drawing electric field lines, they are shown originating from a positive charge and ending on a negative charge. Similarly, as shown in Figure 9. 1 on the next page, magnetic field lines can be shown originating on the north pole of the magnet and then travel back toward the south pole. The difference between electric and magnetic fields is that this motion of the magnetic field lines does not end at the south pole. Instead, the magnetic field lines create a loop by flowing through the bar and returning to the north pole.
Field orientation is always determined by the north and south pole because the field lines always travel from north pole to the south pole in a continuous loop. And similarly to electric field lines, the closer the field lines are to one another, the stronger the magnitude of the electric field. Meaning that the magnetic field of a bar magnet is strongest inside the bar and at the ends with the poles. Figure 9. 1 Part II – The Solenoid Another type of magnet, where an electric current I flows through a conducting object, inducing a magnetic field, is called an electromagnet.
When there is no current (no moving charges within the object), there is no magnetic field. While an electromagnet is functioning, the magnetic field depends upon the current and the direction of its flow. A north-south dipole is created when electrons travel in one direction through a wire, but then an opposite dipole is formed when the electrons flow in the reverse direction. The orientation of the magnetic field lines form closed loops in the wires of electromagnets just as they did with the bar magnet. Field direction depends on the direction of current flow and can be determined by the right hand rule for electric currents.
Using Figure 9. 2 as a reference, imagine the fingers of the right hand curling around a wire and end by touching the palm. The direction of the fingers curled around the wire represents the direction of the magnetic field. In addition, the direction of the current is represented by the perpendicular position of the thumb. In the case of Figure 9. 2 the direction of the current is upward. Figure 9. 2 Reversing the direction of the current will also reverse the direction of the magnetic field. Therefore, imagine the same scenario with Figure 9. as stated earlier, except the current (thumb of right hand) would point downward. This would result with the magnetic field going into the page on the left side of the wire and then coming out of the page on the right side. And as is depicted in Figure 9. 3, the right-hand rule for electric currents can also be applied when considering a loop of wire. Figure 9. 3 Solenoids are composed of a wire that has been coiled multiple times and each coil has been compacted together to create a hollow column of multiple wire loops (Figure 9. 4). Similarly to the single loop in Figure 9. , the magnetic field generated by the current will travel in the same direction inside and outside the column of loops. Looking at Figure 9. 4 and referring back to Figure 9. 1, the column of compact coils looks and displays the same features as a bar magnet with regards to the way the field lines loop around the solenoid. Figure 9. 4 Looking at Figure 9. 4 displayed above, which is a cross-section of a solenoid, the current in the coils is flowing into the page (shown as ? ) at the top and out of the page (shown as ?? ) at the bottom. Rotating the image 90 degrees, having the “north pole” of the solenoid above the lane of the page, will show that the current flows in counter-clockwise direction. The direction of the magnetic field inside the solenoid is uniform and flows in one direction. The magnetic field outside the solenoid is also uniform but flows in the opposite direction. Like a bar magnet, the compacted field lines inside and at the ends of the solenoid make the magnetic field strongest at these regions. From these observations, summing the magnetic fields of each loop will give the total magnetic field of the solenoid. Part III – The Helmholtz Coils
Helmholtz coils and a solenoid both function in a very similar manner. Looking at Figure 9. 5, it shows that these two coils are identical and are made up of wire that has been looped around multiple times. They are placed at a specific distance apart from each other, and this distance is equivalent to the radii of each coil. This creates a strong, uniform magnetic field in the space between them. Figure 9. 5 Using electromagnetic theory equations, it can be seen that the magnitude of the magnetic field is directly proportional to the magnitude of current running through the coils.
This relationship can be expressed by: Where B represents the magnetic field, I represents the current, N represents the number of loops in each coil, R represents the radius of the coils (and the distance between the coils), and K represents a constant (Equation 9. 6) – µ0 is the permeability of free space: 4?? 10-7 Tm/A This allows the equation to be simplified further, because the magnetic field is proportional to the constant value of a specific set of Helmholtz coils multiplied by the current. This constant value is represented by the following equation:
In cases where there is an external magnetic field B0, this field would just be added to the total field created by the Helmholtz coils. Rewriting Equation 9. 8 to include this new parameter produces Equation 9. 9: B = ? I + B0 Equation 9. 9 Part IV – The e/m Ratio of the Electron In the same way that an electrical field can exert an electrical force on charged particles, a magnetic field can exerts a magnetic force on moving electric charges. This is mathematically represented by: A magnetic force F is exerted on the moving electric charge q moving at a elocity v and B represents the magnetic field and ? represents the angle between the magnetic field and velocity vectors. Magnetic force is perpendicular to the velocity and magnetic field vectors. This can be proven by another right-hand rule where pointing the thumb of the right hand in the direction of the velocity and fingers in the direction of the magnetic field, then the perpendicular vector that points outward from the palm of the hand represents the direction of the magnetic force. This can only be used with positive charges though.
When determining the direction of the electric force with negative charges, it possible to either use the left-hand rule (same parameters as right-hand rule) or just invert the right-hand rule by rotating the direction of the magnetic field (fingers) 180 degrees but keeping the direction of the velocity (thumb) the same. Figure 9. 10 further explains this concept. Figure 9. 10 Looking at Figure 9. 10, the circular motion of the charged particle in a uniform magnetic field makes it a centripetal force.
This means that at any point on the circular path, the direction of the force is perpendicular to the travel path and is being directed inward toward the center of the circular path of the charged particle. This relationship between the magnetic force and centripetal force that is observed can be supported mathematically by use of the following equation: Setting ? to equal 90 degrees will results in sin? being equal to one, then by rearranging Equation 9. 13 the charge per unit mass for a given particle in a uniform magnetic field can be determined:
Electrons are often accelerated through electric potentials, so understanding how Equation 9. 14 relates to the potential difference across the two plates is essential. Using the Law of Conservation of Energy, it is observed that K1 + U1 = K2 + U2. Assuming that the initial velocity of the particle and the final electric potential energy at the other plate are both zero, the expression can be simplified to U1 = K2, resulting in the following equation: For a charged particle, the value of e is constant with a value of 1. 602? 10-19 C. This value along with the literature mass of an electron at 9. 09? 10-31 kg makes it possible to calculate the literature value of the charge to mass ratio of an electron, which turns out to be 1. 758? 1011 C/kg. The e/m tube consists of a cathode (negative) plate that lies below a circular anode (positive) plate that stretches across the tube’s diameter with a filament that sits just below the cathode plate. The enclosed tube is filled with low pressure Helium gas. The task of the filament is to strip the electrons away from the Helium atoms by heating them up. Then the voltage potential between the cathode and anode plates accelerates the electrons.
When these accelerated electrons collide with the gas molecules, they produce a blue-green glow. The e/m tube sits inside a set of Helmholtz coils and when there is a current flowing through the coils the path of the accelerated electrons becomes circular due to the uniform magnetic field. Determining the e/m ratio requires the use of Equation 9. 17. By adjusting the accelerating voltage V and current I of the Helmholtz coils, the diameter and radius of the beam path can be calculated with the use of a mirrored scale located behind the e/m tube.
Experimental Setup Part I – The Bar Magnet By laying a bar magnet on the lab bench, it will be possible to view the magnetic field produced by passing a compass over the magnet starting from one end and ending at the other end. Looking at the compass, the north pole of the magnetic portion will point in the direction of the south pole of the bar magnet. The three-dimensional direction of the north pole on the compass should be tangent to the magnetic field lines produced by the bar magnet at every point along the path from one pole to the other.
By passing the compass over and around the bar magnet, the field lines are able to be visualized and sketched with the general rules for mapping a magnetic field as stated and shown earlier in Figure 9. 1. Part II – The Solenoid Figure 9. 7 Using figure 9. 7 as a guide, the solenoid was connected to the circuit. The red wire was attached to one end of the solenoid and then the black wire was attached to the opposite end. After this was done, both wires were the attached to their respectively colored outlets on the power source.
Once this has been completed, the power source was turned on and the power source was set to measure the current in amperes A. Then the 20 V knob on the power source was adjusted until the current read 2. 0 A. As performed in Part I with the bar magnet, by moving the compass around the solenoid, the magnetic field lines of the solenoid can be found and sketched. Other characteristic features that can also be found and should be sketched are the current direction and the position of the electromagnetic north and south poles of the solenoid.
Next, using the right hand rule (explained earlier with Figures 9. 2 and 9. 3), a hypothetical scenario can be sketched displaying the effects of the directional change of the current. Sketching the direction of the current will then make it possible to predict and then sketch the direction of the magnetic field lines and the location of the solenoid’s electromagnetic poles. Part III – The Helmholtz Coils Figure 9. 8 Using Figure 9. 8 as a guide, the Helmholtz coils were set up in the following circuit. In the center, between the two coils, was the e/m tube, which will be used later in Part IV.
It was very important that the circuit wires were connected to the terminals labeled “HELMHOLTZ COILS,” on the apparatus in order not to damage the e/m tube. The same electromagnetic parameters as in Part II of the experiment were used here: the voltage was set to its maximum value and the current was set to 2. 0 A. Then as before in Parts I and II, a compass was passed in and around the area of the Helmholtz coils in order to detect the magnetic field surrounding the coils and the direction of current flow.
Once this information was found, a sketch of the apparatus along with the direction of the current, the direction of the magnetic field, and the location of the coils electromagnetic poles could be sketched. After all the necessary information what sketched out, the power supply was turned off. The next portion of Part III involved the use of a magnetic field sensor (Figure 9. 9) to record the magnitude of the magnetic field around the Helmholtz coils. The Logger Pro™ computer software recorded the data from this sensor. Figure 9. Before collecting data it was important that the bar magnet from Part I be moved as far away as possible so it will not interfere with the sensor’s readings. The magnetic field sensor was set to 6. 4 mT. While using the sensor, the small white tip (see Figure 9. 9) needed to be oriented perpendicular to the direction of the magnetic field, which was determined earlier in this part of the experiment. After setting the zero point for the sensor on the Logger Pro™ screen, the power was turned on, and the current reset to 2. 0 A.
The sensor was placed inside the coils – as close to the center as possible without touching the e/m tube – in the same plane as the axis of the coils. Logger Pro™ then determined the magnetic field of the coils and this reading was recorded as the magnetic field near the e/m tube. The sensor was then placed just outside of the coils, but still along the plane of the axis. And just as before, Logger Pro™ determined the magnetic field of the coils and this reading was recorded as the magnetic field outside the Helmholtz coils.
Continuing with this experiment, the sensor was again placed near the center of the coils along the same plane as the axis of the coils. However, this time the current in the coils was set at 0. 5 A. Logger Pro™ was used to collect the magnitude data of the magnetic field over a time interval of about ten seconds. Then using these data points, the mean magnitude was determined by the Logger Pro™ program and recorded along with the current used. These steps in the procedure were then repeated by increasing the value of the current by intervals of +0. 5 A up to a final reading of 3. A (it was important to remember that that current readings needed to be made from the ammeter, not the power source). Then using the plot function of Logger Pro™, a Magnetic Field versus Current plot was created from the data that was recorded. From this plot it was possible to determine the slope and the intercept of the line of best fit for the data. These two points represent ? and B0 in Equation 9. 9 respectively and were recorded in order to solve Equation 9. 9 and compare the experimental magnitude of the magnetic field with the expected value.
Part IV – The e/m Ratio of the Electron Figure 9. 11 For the final part of the experiment, some extra connections were added to the circuit of the e/m tube apparatus from Part III. The circuit was connected to the “Filament Supply” terminals and the “500 VDC” terminals as shown in Figure 9. 11. Before turning on the power supply, a cloth hood was placed over the apparatus, the e/m MEASURE toggle switch was flipped up, the Volts/Amps. switch was set to amps, the Current Adj. knob was set to five (straight up), and the VOLTAGE MONITOR SELECTOR switch was set to 500 V.
Once all these parameters were completed, the power supply was turned on and the filament was allowed to heat up. After the filament heated up enough, the 500 VDC knob was adjusted to increase the voltage until a visible beam was produced from the filament. This value of the voltage was recorded and acted as the starting point for the remainder of the experiment. Before proceeding with the experiment, the Focus knob was adjusted in order to focus the beam being produced and the 20 VDC knob was used to adjust the current, and in turn the magnetic field, until the beam formed a closed circle.
The value of the current at which the beam formed a circle was recorded for later use. Now proceeding with the experiment, using the mirrored scale in the back of the apparatus, the position of the outside portion for the left side of the beam was found and recorded. Then the same was done for the outside portion of the right side of the beam. These data points were then used to determine the diameter and radius of the circular beam in the apparatus. This process was then repeated sixteen times with each trial being increased by approximately +10 V and the results of each trial were recorded.
Data and Analysis Part I – The Bar Magnet The resulting sketch of the magnetic field around the bar magnet was very similar to the one in Figure 9. 1, and is shown below: The north pole of the magnet and the field lines were observed to be tangent with each other. The north arrow pointed vertically downward toward magnet when placed above the magnets midpoint. In every other location above the magnet, the north arrow would orient itself toward the south magnetic pole. This explains the reasoning behind why the north geographic pole of the Earth is actually its south magnetic pole.
A navigating compass points to what is considered to be the Earth’s geographic north pole, which is counterintuitively the Earth’s magnetic south pole. This characteristic of Earth can be clarified by remembering the concept that opposite poles of magnet attract each other. As stated earlier, magnetic field lines loop from north to south. So when the arrow of a compass, which is a magnet, reads north, this is a result of the magnetic north pole of the arrow being attracted to the magnetic south pole of the Earth.
So in a sense, it is more correct to think of the Earth’s north pole as its “north-loving” pole. Part II – The Solenoid The final sketches of the magnetic field around the solenoid, including both the direction of current flow and location of the north and south magnetic poles of the solenoid, followed the same conventions displayed in Figure 9. 4. In the sketches shown below, one depicts the effects of the current traveling in one direction while the other shows the effects of the current traveling in the opposite direction:
The lines were drawn with the observations made with the same compass used in Part I and the results were the same. Like in Part I with the bar magnet, the electromagnetic components matched perfectly. The magnetic field lines from electromagnetic north pole traveled in a looping path exactly the same as the magnetic field lines of the bar magnet. The image on the right shows the current flowing in the forward direction with the flow of the current going into the page at the top of the solenoid, and the current going out of the page at the bottom.
These observations agree with the right-hand rule for the direction of current flow and magnetic field in the wire loops of an electromagnet. If the current were reversed, the polarity of the solenoid would have changed as well. This is shown in the image on the left, with the current traveling in the reverse direction with respect to the right image. Even though this trial was actually not conducted, the sketch can be proven to be correct because of the looping right-hand rule previously explained in Figures 9. 2 and 9. 3. Part III – The Helmholtz Coils
After sketching the Helmholtz coils and the field they produced, it was observed that the direction of the current flow through the coils and the magnetic field lines in and around the Helmholtz coils exhibited the same electromagnetic characteristics as the solenoid and the same basic magnetic principles as the bar magnet. The sketch provided below confirms these results: For this particular set of Helmholtz coils, the number of turns in the coils was 130 and the radius of each coil was . 15 m. In the region closest to the central axis, the magnetic field was the most uniform and then lost its uniformity at around . 8 m away from the center of the coils. The area surrounding the coils showed the looping pattern of magnetic field lines indicating that each coils created its own electromagnet that was then summed together to create an even larger electromagnet. Using the magnetic field sensor probe and Logger Pro™, the magnitude of the magnetic field near the center of the coils was found to be 1. 655? 10-3 T and the magnetic field just outside of the coils was found to be 1. 135? 10-3 T. These data show that from the center of the Helmholtz coils to the outside edge the magnitude of the magnetic field decreased by 31. 1 %. The magnetic field sensor probe and Logger Pro™ were then used to collect the mean magnitude of the Helmholtz coils near the center over a period of ten seconds. The recorded current and magnetic field values are displayed on the next page in Table 1: Table 1 – Current and Magnetic Field Values for the Helmholtz Coils Current I 0. 50 A 1. 00 A 1. 50 A 2. 00 A 2. 50 A 3. 00 A Magnetic Field B 3. 850? 10-4 T 7. 795? 10-4 T 1. 202? 10-3 T 1. 638? 10-3 T 2. 122? 10-3 T 2. 475? 10-3 T Using these values, the plot graphing function of Logger Pro™ was used first to graph the data and then to determine the ? alue and B0 by creating a line of best fit through the data points. Looking at the Magnetic Field versus Current graph displayed below (please note that the data on the plot is displayed in gauss, not tesla), the ? value (slope) was calculated by Logger Pro™ to be 8. 522? 10-4 T/A and the B0 value (intercept) was found to be -5. 777? 10-5 T. As stated earlier, Equation 9. 8 could determine expected value of ? , and using the given number of loops in the coils and radius of each coil, the expected ? was calculated to be 7. 791? 10-4 T/A.
The percent difference between the expected and observed values of ? was calculated to have moderate agreement of 9. 378%. Part IV – The e/m Ratio of the Electron For the final part of the experiment, the while increasing the voltage, the positions of both the left beam edge and the right beam edge were recorded in order to find the beam diameter and radius which was then used to find the charge to mass ratio of the electrons inside the e/m tube. Table 2 below displays the recorded and calculated results from the experiment. Table 2 – e/m Tube Data
Accelerating Voltage V 116 V 137 V 147 V 157 V 167 V 177 V 187 V 197 V 207 V 218 V 227 V 237 V 247 V 257 V 267 V 277 V Coil Current I 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A 1. 85 A Beam Diameter D 0. 055 m 0. 055 m 0. 065 m 0. 065 m 0. 074 m 0. 075 m 0. 075 m 0. 080 m 0. 081 m 0. 081 m 0. 083 m 0. 086 m 0. 090 m 0. 091 m 0. 093 m 0. 095 m Beam Radius r . 0275 m . 0275 m . 0325 m . 0325 m . 0370 m . 0375 m . 0375 m . 0400 m . 0405 m . 0405 m . 0415 m . 0430 m . 0450 m . 0455 m . 0465 m . 0475 m Magnetic Field B 1. 19? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T 1. 519? 10-3 T Beam Edge Left 0. 040 m 0. 035 m 0. 045 m 0. 045 m 0. 052 m 0. 051 m 0. 050 m 0. 055 m 0. 055 m 0. 055 m 0. 056 m 0. 056 m 0. 060 m 0. 060 m 0. 062 m 0. 063 m Beam Edge Right 0. 015 m 0. 020 m 0. 020 m 0. 020 m 0. 022 m 0. 024 m 0. 025 m 0. 025 m 0. 026 m 0. 026 m 0. 027 m 0. 030 m 0. 030 m 0. 031 m 0. 031 m 0. 032 m e/m 1. 330? 1011 C/kg 1. 570? 011 C/kg 1. 206? 1011 C/kg 1. 288? 1011 C/kg 1. 057? 1011 C/kg 1. 091? 1011 C/kg 1. 153? 1011 C/kg 1. 067? 1011 C/kg 1. 094? 1011 C/kg 1. 152? 1011 C/kg 1. 142? 1011 C/kg 1. 111? 1011 C/kg 1. 057? 1011 C/kg 1. 076? 1011 C/kg 1. 070? 1011 C/kg 1. 064? 1011 C/kg In order to get a more exact value from the various collected data points for the e/m value, the average was taken of these data and was calculated to be 1. 158? 1011 C/kg. This value had an abysmal 34. 17% error in agreement from the literature value 1. 759? 1011 C/kg for e/m. The standard deviation was calculated to be ±1. 70? 1010 C/kg. Adding this value to the calculated average gives 1. 295? 1011 C/kg, and subtracting this value from the calculated average gives 1. 021? 1011 C/kg. The range between these two calculated values in called the standard deviation of the mean. Unfortunately, the literature value of e/m does not fall into this range between these values. This provides further support towards the abysmal agreement that was calculated earlier. Conclusion Each part of the experiment was successful in terms of the observing the nature of the magnetic field on both permanent and electromagnets.
The permanent bar magnet showed how the magnetic field created by these substance travel on a path that leads from a north pole and then ends at a south pole. This observation set up a theme for the rest of the experiment, which was conducted with electromagnets. Using the helpful guidelines established by the right-hand rules of currents through wires and loops of wires, it was possible to determine the direction of the magnetic field of an electromagnet and see that the electromagnet exhibits the same magnetic qualities as the permanent magnet when it has a current flowing through it.
When collecting data within the Helmholtz coils, the determination of the magnetic field went more smoothly than the determination of the charge to mass ratio of the electron in the e/m tube. The moderate 9. 378% difference in the ? value and the abysmal 34. 17% difference in the e/m ratio indicates that there were some sources of error that occurred when collecting data during the experiment. The sources of error were mostly due to errors in procedural protocols and other uncertainties. Many things could have gone wrong when measuring the magnitude of the magnetic field in the Helmholtz coils.
For example, when determining the current traveling through the solenoid circuit and Helmholtz coil circuit, reading the current from the power source instead of the ammeter could lead to bad data. Also the magnetic field sensor may not have been placed in the most efficient position, altering the obtained magnetic field values. Furthermore, reading the beam position values for the e/m tube underneath the cloth hood was very straining on the eyes and it was difficult to focus on the mirrored scale. There was also background light from the laboratory room’s ceiling lights that could have made seeing the beam more difficult.
Therefore, technological, human, and environmental errors were major components of the errors that occurred during the experiment The best way to prevent error is to follow procedural protocol. In this case, even when protocols were followed, there was still error due to external sources such as equipment uncertainty, disruptive light, and human setbacks. The solution to amount of light present is a simple one – turn off the lights. However, the errors brought upon by the equipment can only be overcome by making sure that every switch, dial, lever, etc. is set exactly where it needs to be.
Also, in terms of the magnetic field sensor, placing it precisely at the exact location to get a perfect reading is not probable, so just making sure that the values that are recorded are consistent is the best that one can do. Human errors are going to occur regardless because people are not perfect. The only way to overcome human error is by using logical thinking (knowing when a value is wrong because of our knowledge of physics) and maintaining a strong work ethic (continuing to follow protocol and not getting bored or tired no matter how difficult or tedious the procedure).
Setting all these errors aside, the main purpose of the experiment was executed in the correct manner and as a result a greater understanding of the workings of magnetism in both permanent magnets and electromagnets and their effects on the electron was formed. Works Cited Clark, Russell J. “The Charge/Mass (e/m) Ratio of the Electron. ” Introduction to Laboratory Physics. 3rd ed. Dubuque: Kendall Hunt Pub, 2012. 117-30. Print. Cutnell, John D. and Kenneth W. Johnson. Physics. 8th ed. Hoboken: John Wiley & Sons, 2009. Print.

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