Physics307L:People/Ozaksut/E over m

http://openwetware.org/wiki/Physics307L:People/Ozaksut/Notebook/071121 http://openwetware.org/wiki/Physics307L:People/Ozaksut/Notebook/071128 http://openwetware.org/wiki/Physics307L:People/Ozaksut/Notebook/071121

http://openwetware.org/wiki/Physics307L:People/Ozaksut/Notebook/071128

e/m

• e/m Lab Manual

Goal

• measure the charge to mass ratio of an electron
• understand EM

Theory

Electric currents induce magnetic fields. The magnetic field along the axis of symmetry generated by Helmholtz coils is given by an equation relating number of coils, current in the coils, and radius of the coils. Helmholtz coils are special because the magnetic field between them is nearly constant for a larger area than for other magnetic field inducing coils. We know the number of coils and radius of coils, and we will control the current in the coils using a power supply.

The magnetic field generated by a current in the counterclockwise direction will be parallel to the ground, in the direction of the experimenter. The force on a charged particle in the magnetic field will be qvXB, so any particle with initial velocity in the counterclockwise direction will continue to travel in a circle, in the counterclockwise direction, due to the force on the particle directed towards the center of the circle.

The centripetal force on the electron is ma=mv^2/r, and the centripetal force is equal to the force the electron feels from the field, we equate ma=mv^2/r=qvXB=qvB because v, B are perpendicular always. q/m=v/rB. now, r=mv/qB.

The kinetic energy of the electrons when they leave the electron gun is given by mv^2/2=qV. so the velocity is v=(2qV/m)^1/2. q/m=v^2/2V. v/rB=v^2/2V. v=2V/rB.

We will accelerate electrons off of an electron gun with different voltages between Helmholtz coils and measure the radius of the circular path. With this information, we can calculate the ratio e/m.

Equipment

• Helmholtz coils
• three power supplies
• electron gun and helium bulb
• three multimeters

Setup

We set up the equipment as outlined in Professor Gold's lab manual.

The Procedure

To calibrate the spectrometer, we set up the system as described above, and insert a mercury tube into the light box. When we have rotated the bulb to make sure we have maximum intensity, we remove the cover on the prism and loosen the prism from the rotating platform. We set the knob to read one of the given wavelengths of the Balmer series for mercury and manually rotate the prism until the corresponding colored spectral line lies on our crosshairs. We then fix the prism to the platform, return the cover (which reduces ambient light), and take readings on the knob of the other spectral lines we observe as they pass our crosshairs. Because angles and distances in the spectrometer system are related by real gears, we are careful to reduce any error caused by gear shift by taking measurements in the same direction each time.

Once we take multiple measurements of the wavelengths of the spectral lines we see by reading corresponding wavelengths off the knob, we can calculate, on average, how much our measured values deviate from the exact values we want to read, and take that percentage into consideration when we take measurements of spectral lines of unknown elements.

After we calibrate the spectrometer, we replace the mercury bulb with a hydrogen bulb, and take measurements of hydrogen's spectral lines. We take multiple measurements in order to get the best average values of Balmer line wavelengths and use our calculated error from our calibration in order to better fit the Rydberg equation for quantum states. (The wavelength of a photon in the emission spectrum is related to the energy of the photon by E=hc/lambda, and that energy corresponds to the difference in energy between the Hydrogen atom's electrons' states. When an electron gains energy- in this case, mostly kinetic energy from the heating of the gas in the tube- it can transition into an excited state, and when it falls back down to its ground state, it releases energy in the form of one or more photons of specific wavelengths, depending on which transitions are made, governed by the Rydberg relationship.)

^{SJK 02:06, 13 November 2007 (CST)}

02:06, 13 November 2007 (CST)
You can read up about why Deuterium should be different, and also by how much it should differ. Then you can talk about whether you have enough resolving power in your instrument to be able to measure the difference. What the heck is the orange line, though? That is really funny, and I would like to see it!

Because deuterium (hydrogen with a neutron in the nucleus) is only more massive than hydrogen, but doesn't add any charge to the system, I wouldn't really expect the Balmer lines to be different. If they were different, I would expect they would all be different. This is not what we observed. All the deuterium lines were almost exactly the same, with the addition of an orange Balmer line. Because the relationships between wavelength and state transitions were the same as hydrogen for all the other deuterium lines we observed, I'd guess that the Rydberg constant for deuterium would be the same as for hydrogen, but we wouldn't use the exact same relationship. Or, if the Rydberg constant was different, I'd assume it would affect differences in energy level differently: similar to an index of refraction, for certain wavelengths, the change in wavelength might not be detectable, but for others, it might be be very noticeable. But in this case, because every other line didn't noticeably shift, I would think that our spectrum would be affected at either extreme (say, an IR line would shift into the red, or a UV would shift into the blue), but this, alas!, is not what we observe.

Lastly, we decided the resolving power of our instrument could only be described qualitatively. We were able to resolve three green spectral lines in krypton as our limit. The distance between the edge of each spectral line was about half the width of one of our crosshairs. If we knew the real width of our crosshair, we'd know the real distance between spectral lines at our eye, corresponding to some angular spread achieved by the prism. (distance from eye to prism)sin(theta)

Data

Please see the excel spreadsheet of our data and calculations here: Media:REAL BALMER.xlsx

Calculations

1/{\lambda}=R(1/4-1/n^2)

Because the fraction (1/4-1/n^2) gets bigger as n grows, and I want to keep 1/R constant, I will match increasing n with decreasing lambda.

Error Analysis

^{SJK 02:00, 13 November 2007 (CST)}

02:00, 13 November 2007 (CST)
You have the most important parts, namely the final result, the uncertainty, and the units. Those are all critical and allow the reader to understand everything. Less critical is to tweak it to have the right amount of significant figures and make it easy to read. For example, you don't need so many sig figs on your uncertainty. One way to write it would be: (1.098 +/- 0.004) * 10^7 (1/m), which I think is better. Very precise measurements!!!

I calculated R for hydrogen to be 1.098X10^7 +/- 4091. m^-1. Compared to the real value of 1.09677X10^7, my calculation based on my measurements was, on average, .11% off.

Lab Critique

This lab made me feel like a Victorian drylabscientistphilosopher...minus the ether. A+!

^{SJK 02:00, 13 November 2007 (CST)}

02:00, 13 November 2007 (CST)
I don't know what that is, but I tend to agree :)

Koch Comments

Excellent summary, fun to read, and very nice data! I will email some other comments.