Introduction

In 1897, J.J. Thomson performed the Nobel prize winning experiment to calculate the charge-to-mass ratio of an electron (e/m). The purpose of this experiment is to recreate and confirm his findings (the accepted value for the ratio is 1.758820024(11)× 〖10〗^11 C/kg). To recreate the experiment, we will use the electron gun to generate an electron beam and induce a magnetic force, equivalent to the magnetic part of the Lorentz Force 〖( F〗_(e,mag)=-e ( v × B ) ), that will curve the beam counterclockwise into a circle (set-up shown in Experimental Description and Results). We can equate the magnetic force to the centripetal force required for uniform circular motion (Equation 2) and use the law of conservation of energy to relate
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We picked values for the accelerating voltage and the magnetizing current for each trial and found the radius of the electron beam with each configuration. The error for each measured value was determined by the instrument used and the fluctuation of values. Using these values and Equations 9 and 10, we calculated the charge-to-mass ratio of electrons (e/m). The error for this value was determined by the propagation of uncertainty formula.

Using the values from Figure 2, we can plot V_accel as a function of B^2 R^2/2. The slope of the best fit line is equivalent to the charge-to-mass ratio of electrons (e/m).

Figure 3: Finding the charge-to-mass ratio of electrons (e/m) using the relation between accelerating voltage and B2R2/2. We used multimeters connected to the CRT and a ruler to obtain the data, some of which manipulated as B2R2/2. In this graph, the slope of the best fit line is equivalent to the charge-to-mass ratio of electrons (e/m). Using the Regression tool in Excel to find the slope, we find that our value for (e/m) is (1.8 ± .2) 1011 (C/kg).
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Then, the charge-to-mass ratio of electrons (e/m) was calculated using those quantities and the magnetic field produced by the Hemholtz coils, which was calculated by equation 10. The uncertainties for the accelerating voltage and the magnetizing current were given by the multimeter. However, we need to calculate the uncertainties of the radii of the electron beam, the magnetic field produced by the Hemholtz coils, and the charge-to-mass ratio of electrons (e/m). In order to do so, we used formulas of error propagation. For the error propagation of the radius, we used: δr= √(((δr_1)/2)^2+((δr_2)/2)^2 ) where r_1 is the radius from left side of the beam and r_2 is the radius from the right side. For the error propagation of the magnetic field produced by the Hemholtz coils, we used Equation 12.

δB= √(((8_0 NδI)/(5√5 R_c ))^2+((8_0 IδN)/(5√5 R_c ))^2+((-8_0 INδR_c)/(5√5 〖R_c〗^2 ))^2

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