Lab 1_ Charge to Mass Ratio of the Electron

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Charge to Mass Ratio of the Electron Bassel Marouni & Joannie Naluz PCS 130 - Physics II Section 10 Vladislav Toronov Fatemeh Lalegani Dezaki

Introduction A magnetic field is when the surrounding space of a magnet is affected and magnetism is observed. Uncharged particles do not create a magnetic field, but charged particles do when they are in motion. Depending on the particle's charge, the magnetic field could have different properties. The charge-to-mass ratio is the charge divided by the mass of a particle, and so different particles each have their unique charge-to-mass ratio. That means that scientists could use the charge-to-mass ratio to identify whether a particle is an electron or a proton. The main objective of this experiment is to calculate the charge-to-mass ratio of the particle with the equipment listed below. This will be determined by collecting the motion of the particle in the magnetic field at different radii and currents. Procedure This experiment was conducted using an e/m apparatus; composed of an e/m tube, Helmholtz coils, a high voltage power supply, a low voltage variable power supply, banana cables, a wooden cover, and a meter stick. The power supply was turned on, and the voltage was set to 230V and left running until a circular, green beam formed. While maintaining the voltage, the current was changed which simultaneously changes the diameter of the circle, both these values were collected. This process was repeated until ten trials of different diameters were conducted, and these values are shown in Table 1. A scatter plot was created to compare the variables of the coil current versus the radius of the beam, and a line of best fit was created. Results and Calculations The diameter displayed through each change of current was observed by looking at the measuring stick inside the e/m apparatus ( Table 1) . The r 2 was dividing the diameter by 2, and then squaring it as shown below, 2 = 7.56 x 10 (-4) m 2 ( 0.055𝑚 2 ) 2 = 7. 5625𝑥10 (−4) 𝑚 The magnetic field, B, was determined by using the equation, 8µ𝑁𝐼 𝑎 125 Where μ 0 is 4π x 10 -7 (resistance around a magnetic field), N is 130 (number of coil turns), I is the current (in Amperes), and a is 0.1525 (the average radius of every coil). A sample calculation for the first trial is shown below, = = = 1.76 x 10 -3 T 8µ𝑁𝐼 𝑎 125 8𝑥(4π 𝑥 10 −7 )𝑥130𝑥2.30 𝑎 125 3.005873312𝑥10 −3 1.705001833 Ten trials were performed and data was collected, this can be seen in Table 1.1 .

Current (A) Diameter (m) r 2 (m 2 ) B (T) 1 𝐵 2 2.30 0.055 7. 5625𝑥10 −4 1.76769 x 10 -3 321742.4433 2.18 0.059 8.7025 x 10 -4 1.67546564 x 10 -3 358138.5247 2.11 0.060 9.00 x 10 -4 1.621666292 x 10 -3 382295.4392 1.97 0.065 1.05625 x 10 -3 1.514067581 x 10 -3 438562.5821 1.86 0.070 1.225 x 10 -3 1.429525736 x 10 -3 491969.4545 1.73 0.075 1.40625 x 10 -3 1.320612647 x 10 -3 568685.0629 1.63 0.080 1.60 x 10 -3 1.252756425 x 10 -3 640602.7795 1.55 0.085 1.80625 x 10 -3 1.152843335 x 10 -3 708436.0145 1.46 0.090 2.025 x 10 -3 1.122100847 x 10 -3 798469.4712 1.31 0.095 2.25625 x 10 -3 1.006816513 x 10 -3 991793.9076 Table 1 - Data Collected on Particle’s Properties as Current Change s It was observed that as the current and the radius were inversely proportional, meaning that as the current increased, the radius of the beam decreased. This was the major tell that the observed particle was not a proton, since proton current and proton radius are proportional. For a proton, if the current was increased, then the radius of the beam would increase with it as well.

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Charge to Mass Ratio Figure 2: Graph illustrating the relationship between charge and mass of an electron using observations, recordings, and measurements of radius^2 in meters and the inverse magnetic field (1/B) in Teslas^-2 Determination of the R^2 value

MATLAB's linear regression model can be used to simulate our charge to mass experiment. We have obtained statistical results using the code "mdl=fitlm=(x,y)" and our R^2 was 0.993. In order to determine the e/m value and percent error, Bock & Kelly (n.d.) determined that R^2 should be close to 1, as the closer it is to 1, the stronger the relationship between experimental predictions and observed values. Determination for e/m value and percent error According to the slope of the radius to charge graph, the e/m ratio is 3.8531x10-6. Comparing the theoretical and experimental charge to mass ratio of electrons, we can identify a large difference when we compare the theoretical and experimental charge to mass ratios of electrons. Since we have a distinct difference, we have determined that the percent error is 100%, which implies that the lab experiment was error-free. Discussion 1. It is possible to determine current, vectors, magnetic field, and force by using the right-hand rule in magnetism. We know from the Daedalon EP-20 Manual that the electron is moving in a counterclockwise direction. As a result, we can establish that the current is going counterclockwise, meaning that the magnetic field has left the plane. 2. Due to the fact that the proton is the only pratical known, the electron observations would require a significant difference between qp/m to claim the discovery of another particle. It would have required this value to be outside the range that J.J Thomson had determined to be the charge-to-mass ratio in order to propose a new particle. A particle subjected to exactly the same electric and magnetic fields in a vacuum has the same path when the charge-to-mass ratio is the same. Subsisting e/m = 2 x 10^11 C/Kg and charge of proton e= 1.602 x 10^-19 C to obtain mass m: m = 1.6 x 10^-19 / 2 x 10^11, so m = 8 x 10^-31 kg. It can be said that it is a new elementary particle since its calculated mass is smaller than that of the proton. 3. Here are some points to consider when conducting an experiment in which electrons move in a circular path within a magnetic field ( as in cathode ray tube or similar setup) - Acceleration : Based on the formula below, a charged particle moves in a circular path with centripetal acceleration : a= v^2/r, where : "a" is the centripetal acceleration; "v" is the speed of the electron; "r" is the radius of the circular path.

A circular path cannot accelerate electrons unless the radius of each circle is kept constant. When the radius of a circle is smaller (tight circle), the centripetal acceleration will be higher, whereas when the radius is larger (big circle), the centripetal acceleration will be lower. - Force : As a charged particle moves in a circular path, the magnetic force (Lorentz force) provides centripetal force: F = q x v x B, where : "F" is the magnetic force;"q" is the charge of the electron; "v" is the speed of the electron; "B" is the magnetic field strength. Electrons will always travel at the same speed and, therefore, the force acting on them is dependent on the radius of the circular path. A smaller radius (tight circle) implies a larger magnetic force, and a larger radius (big circle) implies a smaller magnetic force. So, magnetic forces are greater for electrons traveling in tight circles (smaller r) than those traveling in larger circles (bigger r). Based on the charge-to-mass ratio experiment, this analysis is consistent with observations. Increasing orbital radius (narrow circle) results in stronger magnetic force, increasing centripetal acceleration for electrons. In the cathode ray tube, the electron beam exhibits a sharper curve when this occurs, which may be observed experimentally as a sharper curve electron trajectory. A weaker magnetic force is felt by the electrons as their orbits become larger (great circle) and their curvature increases. Conclusion The discovery of the electron was one of the greatest findings, and this experiment recreated such a discovery with the use of an e/m apparatus. By observing the relationship between the coil current vs the radius, along with the radius vs the magnetic field, we could easily see that an electron behaves differently compared to the proton. It was observed that the radius of the beam decreased as the current increased, which differs from a proton’s properties. It was also observed that the proton’s mass was so much lighter than the proton’s since the e/m ratio was much larger. Hence, this experiment led to the conclusion that electrons exist, which drastically changed the way atoms were perceived. In this graph, we can see that the charge-to-mass ratio, e/m, is 2.9954x10-6, which differs considerably from the theoretical ratio of 1.76×1011C/kg. Using this information, it resulted in an error rate of 100%. Our linear fit of the graph yielded an R^2 of 0.993value, which indicated that the charge and mass had a strong correlation. Our investigation of the relationship between charge and mass and its implications for electromagnetism was greatly benefited by this lab.

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References -Bock, T., & Kelly, A. (n.d.). What is R-Squared? Displayr. Retrieved January 27, 2024, from https://www.displayr.com/what-is-r-squared/ -Knight, R. D. (2022). Physics for Scientists and Engineers: A Strategic Approach, 5e . Pearson