PHYS_2&12_Lab 5_e-m ratio

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Tufts University *

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002

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Physics

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Apr 3, 2024

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LABORATORY 5: MAGNETIC FIELDS AND ELECTRON e/m RATIO Goals:  Investigate magnetic fields and their measurement.  Understand the relationship between electric current and magnetic field.  Collect and analyze experimental data to determine the charge-to-mass ratio of the electron. As usual, directions for things you need to DO will be in boxes, in a serif font like this. Explanatory material will not be boxed, and will be in a sans serif font like this. If you’ve already learned about magnetic fields, you may want to skim or skip some of the explanations and go straight to the activities. If not, the explanations will provide you with the background you need. Solar flares (left) and the Northern lights (right) both have their origins in the motion of charged particles in magnetic fields. Introduction (For a video introduction, see Lab5_Video1 on Canvas.) The discovery of the electron is generally credited to the British physicist J. J. Thomson in 1897. Experiments had shown that it is possible to generate beams of some kind that travel in straight lines and can produce electric currents. The nature of the beams, however, was unknown. Thomson showed that they are deflected by electric and magnetic fields like negatively charged particles with a definite charge-to-mass ratio. Since the measured ratio was much too large for a chemical ion, Thomson postulated the existence of a subatomic particle, called the electron. In the final activity of this lab you will perform a similar analysis, involving the deflection of a beam of electrons by a static magnetic field, and measure the charge-to-mass ratio e/m of the electron. The analysis uses many of the physics principles you have studied or will soon encounter, including centripetal acceleration, Newton’s Second Law, the work-energy theorem, electric potential, and the force on a moving charged particle in a magnetic field. 1 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022

Figure 1 shows the equipment that has been used in this laboratory. Measuring and exploring magnetic fields Goals: Explore the magnetic field around a permanent magnet. The magnetic field has a value at every point in space, and since it is a vector, it has both a direction and a magnitude. The Magnaprobe measures the direction of the magnetic field; in essence it is a small bar magnet that can point in any direction (To see it in action, watch Lab 5 – video 2). The Vernier magnetic field sensor (VMFS) measures the strength of the magnetic field at a point. Place one of the cow magnets 1 in the indicated spot on the recording sheet in your Lab Sheets. Use the Magnaprobe to explore the field of the cow magnet. 1 “Cow magnets” are small, cylindrical permanent magnets with rounded ends. Grazing cows often swallow bits of barbed wire, nails and other metal objects that can cause inflammatory “hardware disease.” Farmers actually have calves swallow these magnets to collect the metal and hold it in the cow’s stomach (rumen) where it doesn’t cause problems. The magnet and metal remain in the cow throughout its life. 2 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022 Fig. 1 . Experimental equipment: (1) Cow magnets, (2) Magnaprobe, (3) Vernier magnetic field sensor (VMFS), (4) Computer with Vernier LoggerPro software and LabPro interface, (5) Magnetic shielding tube, (6) 30 cm ruler, (7) 200-turn field coil, (8) multimeter, (9) power supply and leads, (10) e/m apparatus, including Cenco power supply, leads and wooden cover.

 Mark the direction of the field at the locations indicated on the recording sheet. Important note about the Magnaprobe : When the little magnet gets stuck to something (like the cow magnet), do not tug on the plastic handle . Just grasp the little magnet and pull it free. Read the “Notes about the VMFS” sheet at your bench. Position the sensor near the cow magnet at a point where the reading is around 2-4 mT. Observe what happens when you point the sensor in different directions, while keeping it in the same location. Discuss and note on your Lab Sheets: What do your observations tell you about how the sensor reading depends on the orientation of the sensor relative to the field? Carefully zero the sensor as described in the VFMS Notes, and then measure the magnetic field at a spot a meter or so away from any of the permanent magnets or field coils. Does it read zero? You should be able to detect the “geomagnetic field” – that is, the local magnetic field due to the Earth itself. Note on your Lab Sheets: About how strong is the Earth’s magnetic field? The Magnetic Field Due to an Electrical Current Goal:  Test the relationship between I and B for a circular coil. Electrical currents produce magnetic fields. Specifically, the field at the center of a circular coil carrying an electrical current is: (1) where , N is the number of turns of wire in the coil (200 for our field coil), I is the current in the wire, and r c is the radius of the coil (10.5 cm). The field points along the axis of the coil, and its direction is given by a right-hand rule.  Connect the 200-turn Field Coil, power supply and multimeter (use the 10 A scale and input) so you can measure the current through the coil for various applied voltages.  Measure and plot the magnetic field strength at the center of the coil as a function of the current through the coil.  Find the slope of the best-fit line, to at least three significant figures, and estimate the uncertainty in the slope. 3 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022

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 Use the properties of the coil and Eq. 1 to calculate the expected value of the slope of the B vs. I graph. Do the measured and expected slopes agree within the uncertainty? The e/m Ratio of the Electron Goal:  Measure the charge-to-mass ratio e/m of the electron. A charged particle moving in a magnetic field experiences a force perpendicular to both the direction of its motion and the field, with a direction given by a right-hand rule. For an electron (charge – e ) moving perpendicular to a magnetic field B at speed v , the force’s magnitude is: (3) Because the force is perpendicular to the velocity, it does no work on the electron. Thus the electron’s speed does not change, only its direction, and the electron follows a circular path. Using Newton’s 2nd Law and our knowledge of centripetal acceleration, the radius of the path is related to the field strength and to the charge, mass, and speed of the particle by: . (4) We control the electron’s speed by varying its kinetic energy. Electrons are emitted by a hot metal filament, and accelerated to the desired speed by an electric field, which is established by a potential difference Δ V between the cathode and the anode. The kinetic energy of the electrons is equal to the work done by the field: (5) (The lower case v on the left is the speed of the electron. The upper case V on the right is the voltage. Don’t mix them up!) The electrons travel at extremely high speed through a space that contains a very small amount of helium. A few of the electrons collide with helium atoms, causing the atoms to emit green light. This allows you to see the path of the electrons. 4 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022 Fig. 2 . Helmholtz coil arrangement. http://commons.wikimedia.org Fig. 3 . Diagram of the electron gun.

The adjustable current supply controls the magnetic field, using a “Helmholtz coil” arrangement (Fig. 2), two identical circular coils carrying the same current, separated by a distance equal to their radius. This arrangement gives a magnetic field between the coils that is very uniform, with magnitude . Empirically, we have found the magnetic field in the vicinity of the electron orbits to be 1.36 mT and 1.56 mT for coil currents of 1.75 A and 2.00 A, respectively. For other currents, you can use those values and the fact that the field magnitude is proportional to the current. The scale inside the glass bulb allows you to measure the diameter of the electron orbit. Use Eqs. (4) and (5) to derive an expression relating the radius of the electron orbit, r, field strength B, potential difference (voltage) ΔV, and electron charge- to-mass ratio e/m. Give your derivation on Page 3 of the Lab Sheet. (Of course, now we could look up e and m , use Eq. (5) to find v , and plug that into Eq. (4). But in 1897 Thomson couldn’t do that.) Warning! The following activities include electrical shock hazards! Know whether the high voltage is on, make sure there are no exposed electrical connections, and avoid touching the high voltage leads and connections. Warning! You can touch the glass tubes, but do it with care. They are under high vacuum, and while they are not especially fragile, if they break they can implode violently, producing flying glass fragments. Measurement and Analysis: Read and follow the “Instructions for using the Cenco e/m tube” at your bench. Coil currents in the range of 1.5 – 2.0 A are recommended. The current may drift, so you may need to adjust the power supply to keep the current constant. To test the model of electron orbits expressed in Eqs. 1-4, collect measurements of the accelerating voltage V (potential difference Δ V ) required to give various orbital diameters, for two or more values of magnetic field (coil current) . Discuss how to plot and analyze your data to reveal agreement or disagreement with the predicted dependence of the orbit radius (remember that you’re measuring the diameter ) on the accelerating voltage and magnetic field strength. If your data are consistent with the predicted dependence, you can use your measurements to determine the value of the charge-to-mass ratio e/m. That’s pretty cool. You’re measuring a fundamental and universal constant of Nature. Final Comment: Nowadays, you can look up very precise values of both e and m , so why make such a big deal about e/m ? It’s difficult to measure e or m individually, because the forces we 5 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022

can exert on electrons are generally proportional to e, and the response is generally inversely proportional to m . It took Robert Millikan’s Nobel Prize-winning oil drop experiment, using much larger masses, to find a value of e . Combining that value with the e/m ratio allows us to determine m . 6 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022

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DELIVERABLES AND INTERVIEW At least 24 hours before your interview, post to Canvas:  Your Lab Sheets, including enough detail that your TA can understand how you made your measurements, and can follow your thinking.  A computer graph of B vs. I for the field coil, including a best-fit line with the slope indicated.  A computer graph that you used to determine your best estimate of e/m .  A clear statement of your measured value of e/m , and of your estimated uncertainty in that value.  For all graphs, make sure: o Each axis is labeled with the quantity measured and its units (e.g. “Time (s)”). o Data points include error bars to indicate uncertainty. Come to your interview with your laptop, AND with hardcopy printouts of your graphs. Be prepared to explain and discuss:  For each graph: What is being graphed, and what do the results tell you?  How you decided how many points to measure, and which ones? In retrospect, do you think you should have measured more? Could you have measured fewer? Are there regions where you wish you had more data? Where you could get by with less?  How you estimated the uncertainty in the slope of the B vs. I graph.  How you estimated the uncertainty in your value of e / m . 7 Lab #5 PHYS_2&12_Lab 5_e-m ratio.docx v10 Spring 2022

PHYS_2&12_Lab 5_e-m ratio

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