# User:Alex G. Benedict/Notebook/Physics 307L: Junior Lab/e over m

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# e/m Lab Summary

^{SJK 01:22, 18 November 2010 (EST)}### Summary of Activities

In this lab we measured the ratio of electrical charge to mass for the electron by measuring the deflection of an electron beam by a magnetic field. We collected data for the accelerating voltage used, and for the current used in the coils and measured the radius of curvature for each voltage and current. Since the B field produced is proportional to the current we will refer only to the B field. The model used for the calculation of e/m was e/m=2*V/(R^2*B^2) and implicitly assumes the electrons are traveling in a vacuum.

It is assumed by my partner and I that the tube is filled with He gas, and that the collision of the electrons with the gas excite the valence electrons, and the light emitted is evidence of the electrons losing energy. I am assuming that the pressure in the tube is much less than atmospheric pressure which would give mean free path of about a centimeter, which would imply that most electrons will hit quite a few helium atoms. Assuming that the electron goes about 1 centimeters without hitting a He atom, and it travels a path of about ~3cm*2pi as it goes in one loop since Radius of curvature ~3cm. That gives about 18 collisions per electron. Lets round up, to 20 collisions per electron. Then the electron loses about 2 eV on each collision since that is about the order of magnitude of the energy of a photon of visible wavelength. Therefore most electrons should lose about 40 eV of energy. Which is the same as if they had been moving through a medium with some retarding potential of about 40V. So I decided to model this as the original accelerating voltage minus the retarding potential.

To investigate the effect of the earth's magnetic field on the experiment I acted as though the observed e/m was a function of some parameter d, such that e/m=2*v/(R^2*(B+d)^2) and expanded it in a taylor series about 0, which lead a very small difference for d on order of the Earth's magnetic field. When a similar procedure was performed on the voltage a much larger effect was noticed, which further justifies the use of a retarded potential.

### Lab Notebook

My lab notebook is located here: My Lab Notebook.

### Results

^{SJK 01:21, 18 November 2010 (EST)}The average measured e/m was 217349888422.149 +/- 1940461666.18754 using the simple model which gives a relative error of about 0.24 or 24% which is awful, using a model taking into account a retarding potential proportional to the radius of curvature with the constant equal to 1085.714 gives a average value of 177021197987 +/- 2285072584 and a relative error of 0.0065. And fractional errors of 0.0089 and 0.013 respectively. Which does suggest that there is more variance associated with the new model mean compared to the simple model.

Gold also suggested forming least squares fits with for R vs B inverse and R vs Voltage. Which gave confidence intervals of [204691017245,237221836807] and [121206094246, 133676161824], and just of the hell of it I decided to average them, and got a value that is surprisingly close to the accepted value with a simple average, of 173596838647 and a relative error of about 1% which is pretty cool that it is so close given that there is no real reason to average them. The fitted lines and data are plotted below. Additional information is on the spreadsheet in my lab notebook.

### Improvements for Future Labs

In the future it would be cool to make an model for the phenomenon that uses only differences in the B field to rule out the Earth's magnetic field, and only differences in the voltage to account for the heater voltage.

Odd effects were observed in the low voltage, large B field limit that would be more interesting to investigate, pictures of which are below.