| * 1[[#e/mofElectrons(Physics142)-Goal |Goal]] | | * 2[[#e/mofElectrons(Physics142)-GoogleDocTemplate |Google Doc Template]] | | * 3[[#e/mofElectrons(Physics142)-UsingJupyternotebook |Using Jupyter notebook]] | | * 4[[#e/mofElectrons(Physics142)-Introduction |Introduction]] | | * 5[[#e/mofElectrons(Physics142)-Theory |Theory]] | | * 6[[#e/mofElectrons(Physics142)-ExperimentalProcedure |Experimental Procedure]] | | * 6.1[[#e/mofElectrons(Physics142)-Constructingthecircuits |Constructing the circuits]] | | * 6.2[[#e/mofElectrons(Physics142)-TakingandAnalyzingData |Taking and Analyzing Data]] | | * 7[[#e/mofElectrons(Physics142)-Questions |Questions]] | | * 8[[#e/mofElectrons(Physics142)-SubmitYourReport |Submit Your Report]] | ====== Goal ====== ---- Measure the ratio of charge to mass of the electron. ====== Google Doc Template ====== ---- [[https://docs.google.com/document/d/1ZOIVVbdGqmW2pMgAqYvS56h_wVCtQogiHTV_BZNwdRI/copy|Lab Template]] ====== Using Jupyter notebook ====== ---- In order to do data processing and analysis, we will use Jupyter notebooks (which run on the Python programming language). Download the following notebook file: |[[download/attachments/234358728/e_M%20of%20the%20electron%20Notebook.ipynb?version=1&modificationDate=1580773382000&api=v2 |Click here to download the Juypter Notebook]] | ====== Introduction ====== ---- The Millikan Oil Drop experiment, performed between 1900 and 1911 at the University of Chicago, was the first precise measurement of //e//, the charge of the electron. Measurement of the ratio, e/m for the electron thus also yields the mass of the electron. These are important physical quantities! ====== Theory ====== ---- The force on a charge moving in a magnetic field (known as the //Lorentz force//) is {FIXME $\vec F = q \vec v \times \vec B$   (1) where //q// is the charge, //v// is the velocity of the moving charge, and //B// is the magnetic field.\\ The //magnitude// of this force is given by the scalar form of eq.(1), {FIXME $F=qvBsin\phi$   (2) where {FIXME $\phi$ is the angle between the direction of the magnetic field and the direction of motion of the moving charge.\\The //direction// of the force is given by the right-hand rule (if the charge is positive) and is perpendicular to both the velocity and magnetic field. In the special case that {FIXME $\vec v$  is perpendicular to [Math Processing Error]B→ {FIXME $\vec B$  , eq.(2) becomes {FIXME $\mathrm{F=evB}$   (3) where //e// is the charge of the electron. The electron beam will follow a circular trajectory within the field with a centripetal force {FIXME $F=evB=\frac{mv^{2}}{R}$   (4) where //m// is the mass of the electron and //R// is the radius of the circular path as shown in Fig. 1. {FIXME ${/download/thumbnails/234359051/Circular%20e%20traj%20%28142%29.png?version=1&modificationDate=1580841363000&api=v2}$ \\Fig. 1 path of electron in a magnetic field pointing into the page For a non-relativistic electron, accelerated through a potential V, the kinetic energy is {FIXME $KE=\frac{1}{2}mv^{2}=eV$   (5) Eliminating //v// between eqs.(4) and (5) and solving for //e/m// gives {FIXME $frac{e}{m}=\frac{2V}{R^{2}B^{2}}$   (6) Since one can determine all of these quantities on the right side of eq. (6), it is now possible to arrive at a value of e/m. Note that eq.(6) is derived using the following simplifying assumptions: * The magnetic field //B// is assumed to be perfectly uniform over the entire path. * The electrons are assumed to be moving at constant speed along their entire path. **MAGNETIC FIELD**\\ The apparatus uses a pair of Helmholtz coils to produce a magnetic field. These coils have the special geometry with the separation //S// equal to the radius as shown in Fig. 2. {FIXME ${/download/attachments/234359051/worddav27512d074fd174b78c653d4e60c3baab.png?version=1&modificationDate=1580769781000&api=v2}$ Fig 2. Helmholtz coil geometry Using the law of Biot-Savart, it can be derived that the magnetic field (in Tesla) produced by the coils is {FIXME $B= \left( \frac{8}{\sqrt{125}} \right) \frac{\mu_{o}NI}{S}$   (7) where * {FIXME $\mu_{o}=4\pi \times 10^{-7}$  Tm/A is the permeability of free space, * //N// is the number of turns in each coil (N = 130 turns in your apparatus), * //I// is the current (in Amperes) through each coil (they are in series!), and * //S// is the separation (in meters) between the coils and the radius of the coils. A B field map in Helmholtz coils is shown in Fig. 3. {FIXME ${/download/attachments/234359051/worddav44d0f96478e032ecb4f1fe37cabbb35c.png?version=1&modificationDate=1580769781000&api=v2}$ Fig. 3 //B// field produced by Helmholtz coils (from Wikipedia) ====== Experimental Procedure ====== ---- This section will be most meaningful if read in the laboratory, while viewing the apparatus. ===== Constructing the circuits ===== {FIXME ${/download/attachments/234359051/em_wiring_%28142%29.png?version=1&modificationDate=1580841621000&api=v2}$\\Fig. 4 e/m wiring diagram ===== Taking and Analyzing Data ===== According to eq.(6), we have 3 measured quantities to explore: //V,// //R// and //B//. One approach would be to hold //V// fixed while varying //B// and measuring the resulting //R//. Repeating these measurements for several values of //V// would complete the exploration. This method suggests rearranging eq.(6) as {FIXME $\frac{2V}{R^{2}} = \frac{e}{m}B^{2}$   (8) which has the form y = "slope" * x. Plotted appropriately, the data would be expected to fall on straight lines having slope //e/m//.\\ Before taking data, vary //V// and //B// to get a feel for the ranges which can yield useful measurements. ====== Questions ====== ---- **Question 1:** Plot your data in the form of a straight line.\\ **Question 2:** Perform linear fits to each of the straight lines and obtain the value of //e/m//.\\ **Question 3:** Estimate fractional uncertainties for each of the measured quantities.\\ **Question** **4:** Referring to eq.(8), which uncertainty most affects the value of //e/m//?\\ **Question 5:** Make a rough estimate of the uncertainty in your value of //e/m//.\\ **Question 6:** Compare your value of //e/m// with the literature value given inside the back cover of your lab manual.\\ **Question 7:** Is your value for //e/m// consistent with the literature value when your uncertainties are taken into account? Answer using actual numbers and units! ====== Submit Your Report ====== ---- [[https://docs.google.com/forms/d/e/1FAIpQLSdSpRTfhjpdz05W-WIPD758XDoFu6UZ-yzn_yowgKV2_khSzA/viewform|Use this link to submit your report]] .submit {FIXME $ margin: auto; width:50%; display: block; font-size: 200%; text-align: center; align:center; padding: 5px; border: outset; border-color: maroon; border-width: 8px; $ .submit:hover p a {FIXME $ color: white; $ .submit:hover {FIXME $ background-color: maroon; $

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