Charge to Mass Ratio of the Electron Essay Example

we are using a Helmholtz coil assembly to provide the magnetic field, we can determine B in terms of measurable quantities. More specifically, B is given by v 64µ0 N I B= v (II. 6) 125R where N is the number of coils, I is the current through the coils, R is the radius of the coils, and µ0 is the vacuum permeability [1]. When we substitute Equation (II. 6) into (II. ) to get that e 125 R2 V = . m 32 µ0 N r2 I 2 We will use this equation to calculate the ratio of weighted mean values for V, I, and r. e m. Moving charges are subject to a force under the influence of a magnetic field as determined by the Lorentz Force Equation given below F =q E+v? B, (II. 1) where F is the force, E is the electric field vector, v is the velocity, and B is the magnetic field vector. From the above equation, it is immediately obvious that if the applied magnetic field is perpendicular to the direction of motion of the charge, the force causes the particle to move in a circular path [1].

Furthermore, Equation (II. 1) reduces to F = qvB. (II. 2) (II. 7) using B. Error Analysis We know from mechanics that the centripetal force and the Lorentz force balance each other if the particle forms Our experiment requires us to take multiple current measurements at different accelerating potentials. We assumed that the error in the accelerating potential was simply that of the digital multimeter and the error in 2 the current measurements was the standard deviation of the mean.

As

suggested in the lab manual, we took current measurements for the middle peg for each accelerating potential with the assumption that the standard deviation of the mean would be roughly the same for all the pegs in each accelerating potential[2]. To calculate the standard deviation of the mean, we used the formula below [3] 1 n(n1) n 2. Just as before, we need to calculate the uncertainty in each mean for each peg which is given below V Ij RJ =V I j 2 2 RJ 2 + V 2 I 2 j RJ 3 RJ 2 . (II. 14) And finally, we calculate the total uncertainty in our single mean value which is given by the equation below. Vi 2 r2 I = 1 5 j=1 1 2 m = (xi x) . i=1 (II. 8) (II. 15) V I 2 r2 j j V The next task is to calculate I 2 and the associated uncertainties. We obtained this mean value by calculating the weighted mean. From the equation below n i=1 n i=1 wi 2 i 1 2 ? i V I2 .

Method

We get we have that for our weighted average of 5 i=1 Vi 2 Ii, j Vi I2 i, j 2 V I2 = j 5 i=1 1 Vi I2 i, j 2 . (II. 10) Where the index I will be used to represent different accelerating potentials and j will represent the different pegs. However, from the equation, we can see that we need to calculate the uncertainty in I which we have pre2 i, j sent below Vi Ii, j = V 2 Ii, j 2 + 2Vi

Ii, j 3 Ii, j 2 . (II. 11) We then seek the uncertainty in Equation (II. 10) and that Vi I2 = 1 5 i=1 1 Vi I2 i, j 2. The experiment we performed required that we produce an electron beam that would then be detected by the magnetic field produced by the Helmholtz coils. To do so, we used an exciting lament: a cylindrical carbon anode surrounding a tungsten lament placed inside an evacuated glass chamber. The glass chamber contains trace amounts of vapor-phase mercury in a near-vacuum.

Thus, excited electrons will ionize the mercury atoms, causing the blue beam. It should be noted that the outer edge of the beam was sharper than the inside because this edge is created by the electrons with the maximum kinetic energy. The electron-emitting lament was connected to an external power source and a rheostat. We used a Helmholtz coil to provide a near stable, uniform magnetic field as required for the experiment. Our Helmholtz coil apparatus had two coils of equal radius that are parallel and separated by their radius.

The coils were attached to an external power source and a rheostat. The emitted electrons were then exited through a potential difference. Next, we want to calculate I 2 r2 which is ultimately the j I value we will use for calculations. Here, we find that 5 j=1 V 2 2 I j RJ Vi I 2 r2 j j 2 V 2 r2 I = 5 j=1 1 2 .V I 2 r2 j j.

Results

Circuit diagram for the Helmholtz Coils. In our experimental method, we measured the current through the Helmholtz coil required for the

outer edge of the beam to reach the peg. For each accelerating potential, we calculated a mean value and the standard deviation of the mean. We have presented the standard deviations, accelerating potentials, and current through the Helmholtz coil for each peg and accelerating potential.

It is important to note that our radii decrease with peg numbers with peg 1, 2, 3, 4, and 5 having corresponding radii of 11.54, 10.30, 9.02, 7.75, and 6.48 cm [2]. It should be noted that each measurement for the third peg is simply the average of the values obtained.

Our experiment hinges on the uniformity of the magnetic field produced by the Helmholtz coils, we need to accurately determine the local ambient magnetic field and properly take it into account. We used a compass that the dip angle was roughly 136. 5. We set the apparatus perpendicular to the local field to make the computations simple as the two fields would be coaxial. We then measured the detection of the beam caused by the local field and found that the current through the Helmholtz coils required to counter the local field was -0. 1096 A. To do this, we set the lament current to 3A and the accelerating potential to 20V and slowly increased the current through the Helmholtz coils until the beam was straight. After the equipment setup and the preliminary measurements, we proceeded with the procedure to determine the charge to mass ratio of the electron.

We choose accelerating potentials (24. 985V, 40. 013V, 55. 01V, 70. 03V, and 85. 01V) and the center peg to perform our calibration calculations. For these potentials on the center peg, we did

this by measuring the current necessary through the Helmholtz coil such that the beam of electrons would curve with the outer edge touching this pre-calibrated peg. Standard deviation increased slightly as we increased both our accelerating potential and our current through the Helmholtz coils. We observed with the Pasco Model SF-9584 power supply that the current would drift more significantly as it increased. It's noteworthy that later experimenters replaced this power supply due to this observed instability.

Therefore, there appears to be a systematic trend in the error. The multiple current values were then consolidated such that there were 25 measurements: a measurement for each of the accelerating potentials and for each 4 of the pegs. We then used these values to compute V/I 2 using Equation (II. 10) with the associated uncertainty found via (II. 12). Then, we calculated the average V /I 2 r2 using Equation (II. 13) with the associated uncertainty found via (II. 15). For this value, we got that 1920. 68 ± 2. 4 [ V A 2 M 2 ]. The radii of the Helmholtz coil was found to be. 334 ± . 001m. Combined with the calculated value of V /r2 I 2 . the charge to mass ratio for an electron was found to be 6. 54341±. 00474661e7.

If we compare this with the accepted value of h, we have a percent difference of 62. 7966 %. We also sought to the amount our Helmholtz coils vary from a true idealized system. We found that the separation of the coils was within one millimeter of the radii of the coils. Such a small difference can be neglected. We then,

based on observation, determined that the magnetic field from the lament could be neglected because the radius of the beam didn’t change when the current was swapped. We did observe that at lower accelerating potentials and currents, there was dispersion in the electron beam.

Furthermore, the electron beam became roughly elliptical with a relatively small eccentricity. This error is astronomical and indicative of either poor experimental design or experimental execution. Because we are replicating an experiment that has been well replicated throughout the years, it's safe to assume the experimental design is sound. Therefore, we had execution issues. Because the experiment hinges on direct observation, we can conclude that the difference could arise from both parallax and the possible color-blindness of the observer. Other sources of error may include the nature of the beam at different accelerating potentials and disagreements on what exactly constitutes touching the edge of the peg. We cannot attribute an error this large to things such as slight asymmetries in the coils of the magnetic field in the lament. We can discount the former because variations of less than a centimeter in the height of the coil apparatus as compared to the radius cause a less than variation in the uniformity of the coils.

We can also discount the latter because we saw no discernible difference in the radius of the beam when we switched the current. Finally, there could be an arithmetic error. This seems like the most accurate source of such a large error. While the authors have check their work in multiple different ways, and we feel confident in the correctness of the calculations, our calculations are not beyond reproach.

While our approach to data analysis appears entirely valid, there could easily be a bug in the Mathematica notebook used to do the computations.

Conclusions

The experiment conducted was a direct reproduction of that done by KT Bainbridge from the early 1930s. In that experiment and others like it, the exact and accepted value of the ratio of an electron’s mass to its charge has been determined to be 1. 7588202e8.

References

1. Fundamentals of Physics 8th Edition, David Halliday, Jearl Walker, and Robert Resnick. 2007
2. Determination of the Ratio of Charge to Mass for the Electron: Bainbridge Method, Rebecca Forrest.
3. UH PHYS 3113 Lab Manual, 2010.
4. Data Analysis Presentation, Rebecca Forrest. http://phys. uh. edu/rforrest/ErrorAnalysisLecture. pdf, 2010.