Physics 2 Lab 8

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Massachusetts Institute of Technology *

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8.02

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Chemistry

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Oct 30, 2023

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Measurement of e/m for an Electron Lab Number and Title: 212 Measurement of e/m for an Electron Name: Aaron Hsu Group ID: N/A Date of Experiment: 11/8/22 Date of Submission: 11/15/22 Course and Section Number: PHYS 121A013 Instructor’s Name: Matias Daniel de Almeida Partner’s Names: Paul Svorec, Alex Ack, and Noah Francois 1. Introduction: The goals and objectives for this lab are to measure the charge to mass ratio, e/m, for an electron. When an electron is accelerated from rest through a potential difference V, from the principle of conservation of energy we find these two equations, and ?𝑉 = 1 2 𝑚𝑣 2 with v and m being the electron’s velocity and mass. We will look at two cases ? 𝑚 = 𝑣 2 2𝑉 of both E and B are on and E is off and B is on and when the electron enters a region where we can apply crossead electric fields, the ratio of e/m can be measured. With the first case of both E and B being on, Electron in motion in the presence of crossed electric and magnetic fields. E being in the plus y-direction, B being in the plus z-direction, and the electron moving with velocity v in the plus x-direction, it will experience two forces of electrostatic being and magnetic being where and if these ? ? ? 𝐵 ? ? =− ??𝑗 ? 𝐵 = ?𝑣𝐵𝑗 two forces have the same magnitude and are opposite in direction, the electron will move undeflected through the region of the crossed E and B fields which means, , |? ? | = |? 𝐵 | , and . Two coils in Helmholtz arrangement, which is the distance ?? = ?𝑣𝐵 𝑣 = ? 𝐵 between two coils equaling the radius of the coils, will produce a uniform magnetic field between the coils. The area that is between the coils has a magnitude of net magnetic field being, where N(=320) is the number of turns, I being |𝐵| = 8. 992 × 10 −7 𝑁𝐼/𝑅 the current in the turns, and R(=6.8) being the radius of the coils. With two finite parallel plates being top and bottom, it will produce electric field E over the electron’s path and if we apply a potential difference to the plates and they are also separated by a distance 𝑉 𝑝

d(=5.2cm), the average electric field E in the area between the plates will be . ? = 0.77𝑉 𝑝 ? The second case where E is off and B is on is an electron in motion in the presence of a B-field only. When the electron enters a region of uniform magnetic field B only, the force on the electron is given by the equation . The force will always be |? ? | = |? 𝐵 | ? 𝐵 perpendicular to the velocity and B-field vectors which means the electron moves in a circular orbit. The radius of the orbit can be determined by applying Newton’s Second Law which means, where r is the radius ofthe orbit and when we solve this ?𝑣𝐵 = 𝑚𝑣 2 𝑟 equation for e/m we get, which tells us that if B is produced by a set of ? 𝑚 = 𝑣 𝑟𝐵 Helmholtz Coils and we know v, we can find e/m by measuring for r. An equation for a circle with a center at a,b is and when the electron enters the (? − ?) 2 + (? − ?) 2 = 𝑟 2 region of uniform B at the point (0,0), with initial velocity (vi + 0j) which means that a=0 and b=+- r depending on the direction of magnetic field B. Then the equation of the circle is . ? 2 + (? ± 𝑟) 2 = 𝑟 2 2. Experimental Procedure: The equipment that we used for this lab are a magnetic compass, one set of connecting wires, an old style ammeter, one KV power unit (813) for the tube, electron e/m deflection tube, one stand holder for holding the tube and helmholtz coils, pair of helmholtz coil, and one L.T. power unit (800) for the coils. We followed the same procedure as in the lab manual and we did not have to set up the experiment because the professor already set it up for us. For the procedure with the first step being E and B both on, we would set the accelerating potential V to about 3000 V and then we would adjust the helmholtz coil current I for zero deflection. Zero deflection refers to the electron entering and leaving the plate region at y = 0 cm and we would record V and I. Then, for the second step with E off and B on, we would first turn off the high voltage power supply. Then, on the panel of the power supply 813 we would connect the bottom black cable(-) to the top of the red cable(+) which means moving the black cable from negative to positive. Then we would turn the power supply on and set the output back to the same as the first step at 3000 V and keep the same current I in the helmholtz coils. Finally, we

would measure 6 points both x and y on the path of the electron beam using the centimeter graticule on the mica sheet that is mounted in the tube. 3. Results: Case 1: Points - (2, -0.2) (4, -0.3) (6, -0.3) (8, -0.2) (10, -0.1) Case 2: Points - (2, -0.2) (4, -0.4) (6, -0.9) (8, -1.6) (10, -2.5) Calculation: 3. (5.5 * 10 -5 T) (44423.1 N/m) (8.08*10 8 m/s )m/s 8. e/m = 1.47*10 14 4. Analysis and Discussion:

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We were able to determine an electron's charge to mass ratio by observing the impact of electric fields on the electron's route. The equation for a circle, Newton's second law, and other equations were used to accomplish this, along with the energy conservation principle. The resistance in the cables, which we thought to be zero, is likely to have caused a little amount of mistake. The reading of the graph itself is believed to be the main cause of inaccuracy. The spots that made up the line that fluoresced on the mica were difficult to identify. The goal of this lab, which was to determine an electron's charge mass ratio, was accomplished. 5. Conclusion: From the experiment we have learned how an electron functions when going through two cases of E and B both being on and E off and B on. We have also achieved the objective and learned how to measure the charge to mass ratio e/m for an electron. The experiment has not raised any new questions. Our results were pretty fair with the experiment and we were able to get good points from the graphs in our experiment. We were able to use the points that we got from the experiment in order to plug them in and use them with the equations in the lab manual so that we could complete the goal and objective of the lab.