The Charge/Mass (e/m) Ratio of the Electron
Abstract
This experiment aims to provide insight on the charge/mass ratio of the electron. This study uses the application of magnetic field laws to further understand the properties of an electron. The first component of the experiment uses the properties of magnetic fields to understand electron phenomenon. The first plot analyzed the relationship between current and magnetic field in a solenoid, where a slope of 8.170×10-4 T/A and an intercept of 3.716×10-5 T was obtained. The second component of the experiment used Helmholtz coils and an e/m tube to analyze the effects of a magnetic field on the circular motion trajectory of an electron beam. Here, the mean value for the e/m ratio was calculated to be 3.038 x 1011 C/kg +/- 1.927 x 1011 C/kg. The literature e/m value fell within this range. Generally, the experiment as a whole successfully demonstrated the ability to find the e/m ratio of the electron given the properties of a magnetic field.
Introduction and Theory
J.J. Thompson, a British physicist, built a device in 1897 that studied cathode rays. He was able to measure the bending of the cathode ray, and using his knowledge of electric and magnetic fields, was able to measure the ratio of charge to mass of such particles, which was consistent among other gases as well. This helped him conclude that the cathode ray particles are homogenous, negatively-charged parts of atoms called electrons. This lead to great interest in the properties of electrons, and charges in general, as observed through known properties of electric and magnetic fields.1
Electric fields are created by electric charges. Magnetic fields have two poles, known as north and south. When a magnet is cut in half, the poles cannot be isolated. This results in two magnets, since north and south poles will be instantly adopted by each half of the original magnet. 1
A compass is a unique device that can be used to visualize the directionality of a magnetic field line. Next to a powerful magnet, a compass is able to freely rotate around a set midpoint. This can be seen in Figure 1 below.
Figure 1. Magnetic Field Lines Around a Bar Magnet
Magnetic fields are generated by electric charges in motion. In order to deduce the direction of the magnetic field generated by the current in a wire, the right hand rule can be applied. This rule is applied in two parts. The first part is to point the thumb of the right hand in the direction of the current in the wire. The second part is to twist your other fingers around the wire, and this directionality is the same as the directionality of the magnetic field lines. This can be seen in Figure 2 below.
Figure 2. Right Hand Rule 1: Straight Current-Carrying Wire
Below in Figure 3, the same right hand rule can be applied for a continuous loop of current-carrying wire. Here, it is observed that the direction of the magnetic field is the same throughout the inside of the loop of wire.
Figure 3. Right Hand Rule 1: Current Carrying Loop of Wire
Within a solenoid, just like the loop of wire, the direction and magnitude of the magnetic field is uniform. Since such properties are inherent to solenoids, they are commonly used in creating uniform magnetic regions. This can be seen in Figure 4 below.
Figure 4. Magnetic Field Lines and Polarity of Current-Carrying Solenoid
Helmholtz coils can be described as two thin coils that are separated by a certain distance that is equal to their radii. As long as the current is the same in each coil, they will produce identical magnetic fields. The key difference between a solenoid’s magnetic field and the magnetic field generated by Helmholtz coils is that the latter is easily accessible, making it useful in experimental applications. 1
Picture a thin coil with a certain radius R at a point in space. Given that this point is the origin, and the axis of the thin coil runs along the x-axis, the current of the coil is I, then the magnitude of the magnetic field along the x-axis can be quantified using Equation 1 below. 1
Equation 1. Magnetic Field along Thin-Coil Axis
In this equation, N is the number of turns in the coil, and 0 = 4 x 10-7 Tm/A represents the permeability of free space. This expression can be used to calculate the magnitude of the magnetic field along the center of the Helmholtz coils, where x = 0. The first coil would then be at x = -R/2 and the second coil at x = R/2. Since there is little change in the region between the coils, it is acceptable to calculate the magnitude of the magnetic field at the center where x = 0. This can be seen in Equation 2 below. 1
Equation 2. Magnetic Field at Center of Helmholtz Coils
Below, Figure 5 shows a diagram of when these equations would be applied.
Figure 5. Side view of Helmholts Coils
Since everything except the current I is constant, Equation 5 stated previously can be rewritten as Equation 3 below. 1
Equation 3. Magnetic Field of Helmholtz Coils
Here, the quantity K is equal to the permeability of free space, divided by the constant (5/4)3/2 for a value of 8.99 x 10-7 Tm/A. The quantity (KN/R) will be referred to as “.” In the presence of an external magnetic field 0, the magnitude of such will be added to the field in the center of the coils such that:
Equation 4. Magnetic Field as a Function of Current
To analyze the magnetic field in this experiment, a graph of B vs. I will be plotted. Here, the quantity “” will be the slope, and the quantity “0” will be the intercept. 1
Moving charges can produce a magnetic field. This field exerts a force, F, such that the charge (q), velocity (v), magnitude of the field (B), and the sine of the angle () between the velocity and magnetic field vectors are all directly related to the magnitude of the force. This relationship can be described using Equation 5 below. 1
Equation 5. Magnetic Force of a Moving Charge
The right hand rule for magnetic force is different than the right hand rule stated previously. First, point the palm of the right hand in the direction of the velocity vector. Then, point the fingers in the direction of the magnetic field. Lastly, the direction in which the thumb points is the direction of the force generated by a moving charge in a magnetic field. This is true for all positive charges. For negative charges, the direction of the force is reversed. Since the velocity is perpendicular to the force, the moving charge will travel in a circular path in a magnetic field. This is represented in Figure 6 below. 1
Figure 6. Right Hand Rule 2: Moving Point Charge and Magnetic Force
Circular motion can be related to the centripetal force. Setting this equal to the magnetic force and simplifying the equation such that q = e where e is the charge of an electron, sin = 1, and v is canceled from both sides, solving for the quantity e/m yields Equation 6 below. 1
Equation 6. e/m Ratio of Electron related to Magnetic Field
Since the potential energy of an electron is related to the voltage potential where , and assuming all potential energy is converted to kinetic energy where, setting and solving for v2 yields:
Equation 7. e/m Ratio Adjusted for Voltage Potential
The charge of an electron, e, is equal to 1.602 x 10-19 Coulombs and the mass is 9.1x1031kg. Therefore, the expected e/m ratio is approximately 1.759 x 1011 Coulombs/kg. This experiment uses a glass tube filled with Helium gas that contains a filament that heats up and strips electrons from Helium. The electrons are accelerated based on a given potential difference, V, and emit light due to excitation. These beams bend in the presence of a magnetic field. The radius of curvature in the bend relates to the e/m ratio as seen in Equation 7. 1
A Hall Probe will be used in the latter half of Experiment 1. This wand-like device is used to measure the magnetic field of Helmholtz coils via a copper detector plate that defects negative charges to produce a specific electric potential. This depends on the orientation of the probe and can be used to relate a measured potential difference to the strength of the magnetic field. 1
Experimental Setup and Procedure
Experiment 1: Bar Magnet
Using a compass, map the magnetic field lines of a bar magnet, and sketch this in the lab notebook. Then, using the same compass, sketch the magnetic field lines of a solenoid with a particular current flow. Afterwards, reverse the current flow and sketch again. You should be able to see the Right Hand Rule being followed. Some iron magnets, when heated, can fixate and create a constant magnetic field. Sketch what the iron orbits must look like in the atom for iron to produce a permanent magnetic field. 1
Figure 7. Circuit Diagram of Solenoid Producing a Magnetic Field
Construct the circuit as seen in Figure 7 above. Turn on the power supply, and then set the Volts/Amps switch to “A” to display Amps. Set the 20V knob to 2.0A. Sketch the magnetic fields of the solenoid using a compass, confirming with the right hand rule. Then, turn the power off.
Figure 8. Circuit Diagram for Helmholtz Coils
Construct the circuit above precisely as seen in Figure 8 using the e/m apparatus to avoid damage to the e/m tube. Record the radius and number of turns of the Helmholtz coils. Set the Volts/Amps to “A” to display readings in amps. Set the current adjustment knob to 5. Turn on the power supply. Set the current to 2.0A. Using a compass once again, map the magnetic field of the Helmholtz coils. Use the Hall Probe (set at 6.4mT) to confirm the presence of a universal field between the coils, and test the relationship between B and I, which should be linear using the LoggerPro software. Set the zero point for the magnetic sensor by removing the bar magnet and placing it at a distance of a meter or two away. Place the sensor next to the e/m tube. Set the current to 0.5A. Collect approx. 10 data points at various currents starting with 0.5A and record current and mean magnetic field magnitude in the lab notebook. Be sure to read the values from the analog ammeter and not the power supply. Then, turn the power supply off. Analyze data points on a graph of magnetic field versus current. The slope represents the value of and the intercept will be B0, which is the baseline magnetic field. 1
Experiment 2: Helmholtz Coils
Figure 9. Circuit Diagram for e/m Apparatus
Construct the circuit above as seen in Figure 9. Place a dark cloth over the Helmholtz coils. Flip the switch to the e/m MEASURE up. Set the Volts/Amps to “A” to display readings in amps. Set the current adjustment knob to 5. Set the Voltage monitor selector to 500V. Turn on the power, being sure not to exceed 300V. Adjust the focus knob in the Helmholtz coil current and the 20VDC knob to adjust the current and magnitude of the magnetic field until a circular beam appears. The electron beam must hit the outer-most ring within the scale. Read current from the analog ammeter. Record current in the lab notebook, and calculate the magnitude of the magnetic field given Equation 4. Remove parallax errors by lining up the reflection of the beam with the beam itself. Record positions of the left and right beam radii. Increase the voltage by 10V for each subsequent data point, and record radii values. The e/m ratio will be calculated for each data point, and the mean and standard deviation will be compared to the literature value. 1
Data, Analysis, and Results
A bar magnet has a north and south pole. Field lines were mapped using a compass, and this sketch was obtained:
Figure 10. Sketch of Bar Magnet Magnetic Fields Using a Compass3
In this figure, notice that the compass needle points to the south pole. Thus, it was observed that “north” on a compass actually points to a south magnetic pole. Thus, in the real world, the north geographic pole is actually a south magnetic pole.
A solenoid can have a current run through it, which induces the creation of a magnetic field. This appears as such:
Figure 11. Diagram of Solenoid Magnetic Field5
In this figure, the current travels counterclockwise and induces a uniform magnetic field inside the coil. The north end of the solenoid is on the left, and the south end on the right. If the current was clockwise, then the north and south ends of the solenoid would switch, reversing the direction of the magnetic field lines.
For a bar of iron, it can be subjected to ferromagnetism and be made into a permanent magnet by melting the iron and then rapidly cooling it. The orbitals must be unidirectional for this to be a permanent magnet, and create a magnetic field. The electron orbitals for iron, as such, would then appear as in Figure 12 below.
Figure 12. Orbitals of Ferromagnetic (Permanent Magnet) Iron4
Here, it is observed that, in the 3d orbital, all the spins of the electrons point up. This massive polarization of iron is the main contributor to the establishment of a permanent magnetic field.
The magnetic field near the center of the Helmholtz coils for the third part of Exp. 1 was measured to be 1.650 x 10-3 T. The magnetic field just outside the coils was measured to be 1.300 x 10-3 T. The figure below shows the plot of the measured magnetic field versus the current running through the coils.
Figure 13. Magnetic Field versus Current
The slope of the line was calculated to be 8.170×10-4 T/A, and the intercept was 3.716×10-5 T. Comparing the slope measured via the experiment to the expected value of , where =KN/R, an expected value of 7.908×10-4 T/A was obtained. When compared to the measured slope, this accounted for an approx. 4% difference between the values.
Some of the difference observed could be accounted for due to resistance of the wire or coil. Since Ohm’s law shows that V=IR, and the current, I, determines the magnitude of the magnetic field, some resistance can change the current in the coil, thus reflected in the magnetic field calculations. Additionally, a difference of 4% within the lab environment is quite negligible, as there are many factors with human error that can be associated to the overall procedure. Thus, it is within an acceptable range of experimental uncertainty.
Lastly, the theoretical ratio between the magnetic field at the edge of the coils and the magnetic field at the center of the coil was calculated using Equation 3 and a value of 0.9458 was obtained. This was compared to the measured ratio of 0.7879 and found a 16.7% difference. This can be accounted for, since the measured values were approximations made to only 3 significant figures whereas the theoretical ratio is an exact amount, as well as some human error in lab technique, such as holding the probe unsteadily while measuring/approximating the fields. If lab technique and more accurate measuring was performed, then the uncertainty would have been within an acceptable 10% uncertainty.
In Experiment 2, a table of values were constructed with measured values of the accelerating voltage, coil current, and left and right radii.
Table 1. Derived e/m Ratios from the Electron Beam at Various Currents
From this table, the magnetic field (B) was calculated using Equation 4 using the slope, , and intercept, B0, value from Experiment 1. The beam diameter was calculated by adding the left edge and right edge beam radii. Then, the beam diameter was divided by 2 to find the beam radius. The e/m ratio was then calculated using Equation 7.
The average value of the e/m ratio was 3.038 x 1011 C/kg with a standard deviation of 1.927 x 1011 C/kg. Compared to a literature value of 1.7588 x 1011 C/kg, the experimental value was much higher. However, with the standard deviation, the literature value falls within the range of 1 standard deviation. Of course, there is some error and inconsistency across the 16 data points measured in this experiment. If more data points were measured, it is possible that the mean would be closer to that of the literature value, and that the standard deviation would decrease to account for less variability. Additionally, it was quite difficult to measure and read the values for the left and right radii on the scale, since it was almost impossible to avoid parallax error and there were such little change across various currents, where the radii would change by approximately a millimeter or two. This severely impacts the calculation of the e/m ratio, since the radius is inversely proportional to the ratio. Thus, since the literature value falls within one standard deviation, despite the standard deviation being high, good experimental protocol was used.
Conclusions
Overall, the experiment was a success in its ability to test the properties of a magnetic field, and apply such properties to analyzing the properties of electrons, namely the e/m ratio. Using devices such as the e/m tube and Helmholtz coils to construct a circuit, a uniform magnetic field was created to study electrons and derive the e/m ratio. The comparison of the e/m ratio to the literature value helped facilitate the understanding of the different properties of magnetic fields and the electron.
From Experiment 1, the plot yielded a slope of 8.170×10-4 T/A, and an intercept of 3.716×10-5 T. This was used in Experiment 2 to determine the values of the magnetic field based on the current in the Helmholtz coil. There was, however, abysmal agreement (>25%) between the mean measured e/m ratio and the literature value. However, the literature value did fall within one standard deviation of the mean. Thus, the results obtained supported the experimental theory.
To improve this experiment, improving experimental technique and better accounting for human error could have been done. Better lab technique such as increasing the number of measured data points or having more accurate readings of the values with better equipment could be used. Additionally, it was quite difficult to read the radii of the beam given that the experiment did not take place in a very dark room.
References
1. Clark, Russell J. “The Charge/Mass (e/m) Ratio of the Electron.” Introduction to Laboratory Physics. Kendall/Hunt Publishing Company, 2007.
2. Clark, Russell J. “The Charge/Mass (e/m) Ratio of the Electron.” Lecture Notes. 2016.
3. “File:Electron Configuration Iron.svg.” – Wikimedia Commons. N.p., n.d. Web. 12 Nov. 2016.
4. “Physics – Electromagnetism.” Jom:enjoy:. N.p., n.d. Web. 12 Nov. 2016.
5. “Solenoid.” S as Magnetic Field Sources. N.p., n.d. Web. 12 Nov. 2016.