User:Cristhian Carrillo/Notebook/Physics 307L/2010/11/10

e/m Ratio

 * Please note that Ginny was my lab partner for this lab.

Purpose
The purpose of this lab is to measure the charge-to-mass ratio of the electron by studying the effects of electric and magnetic fields on a charged particle.

Equipment

 * Hewlett Packard DC Power Supply (Model 6384A, 4-5.5V, 0-8A)
 * e/m experimental apparatus (Model TG-13)
 * SOAR corporation DC Power Supply (Model 7403, 0-36V, 3A)
 * Gelman Instrument Company Deluxe Regulated Power Supply (500V, 100mA)
 * 2 BK Precision Digital Multimeter (Model 2831B)

Safety

 * Make sure to ground all power supplies properly before use
 * Check cords, cables and machinery for any possible electrocution points on fuses of cords
 * Protective grounding conductor must be connected to ground
 * Be careful with the mercury tube

Setup

 * We followed the descriptions in | Professor Gold's manual and Alex Andrego's Notebook for the setup.


 * Below are the steps we followed to setup the experiment


 * We first used BNC cables to conncect a regulated 6-9V DC supply rated at 2A to the Helmholtz Coil jacks.
 * Connected the ammeter in series between the supply and the coil jacks.
 * We connected the 6.3V power supply rated at 1.5 A to the heater jacks of the electron gun.
 * Connected a high voltage source at 150-300V DC rated at 40mA to the electrode jacks of the electron gun
 * Please note that the voltage determines the average velocity of the electron in the beam.


 * We then connected the DC voltmeter at the jacks labeled "voltmeter" on the base panel.
 * Be sure to turn the current adjust control to zero and set the switch on the panel to the e/m position.
 * Make sure that nothing is connected to the jacks labeled "Deflection Plates" at this time.


 * Allowed the electron gun filament to heat up for a few minutes after we turned on the heater supply.
 * We then applied a 200V DC potential from the high voltage supply to the electrodes.
 * Then we turned off the light to begin the experiment.
 * Make sure that a black cloth to cover the tube and to backdrop the beam while observing the beam of electrons.
 * We then adjusted the current control until the beam formed a circle by turning on the coil current and increasing the current adjustment control.
 * We then used the scale behind the bulb to measure the radius of the loop of the beam.

Calculations and Analysis
Below is our raw data


 * The following equations were used to calculate the e/m ratio.
 * We found the Helmholtz configuration from Professor Gold's Manual to be:
 * $$x=R/2\,\!$$, $$N=130\,\!$$, and $$R=0.15 m\,\!$$
 * The permeability of free space is given as
 * $$\mu=4\pi\times10^{-7}\frac{weber}{amp-meter}\,\!$$
 * From these values we can calculate:
 * $$B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$$
 * We know that the energy of the electron is equal to the kinetic energy:
 * $${e}{V}=\frac{1}{2}{m}{v}^{2}\,\!$$
 * The magnetic force for a charge is...
 * $${F}_{B}={q}{v}{B}\,\!$$
 * The centripetal force is...
 * $${F}_{c}={m}\frac{v^2}{r}\,\!$$
 * Finally we set the centripetal force equal to the magnetic force and obtained:
 * $$\frac{e}{m}=\frac{{r}^{2}}\frac{{(R^2+x^2)}^{3}}{{({u}{R}^{2}{N}{I})}^{2}}\,\!$$
 * According to Professor Gold's manual, the current accepted value of $$\frac{e}{m}\,\!$$ is:
 * $$\frac{e}{m}=1.76\times10^{11}\frac{C}{kg}\,\!$$

Below is the average and SEM that we found using the above equation $$\frac{e}{m}$$ we obtained when we set the centripetal force equal to the magnetic force.


 * Average
 * $$\frac{e}{m}=1.54\times10^{11}\frac{C}{kg}$$


 * SEM
 * $$\frac{e}{m}=2.62\times10^{9}\frac{C}{kg}$$


 * Percent Error
 * $$\% error=12.4%\,\!$$


 * The other way to find $$\frac{e}{m}\,\!$$ is to plot
 * $$r\,\!$$ vs. $${I}^{-1}\,\!$$, where $$V\,\!$$ is constant.


 * [[Image:Rvs.I]]


 * $${r}^{2}\,\!$$ vs. $$V\,\!$$, where $$I\,\!$$ is constant.


 * [[Image:R^2vs.V.jpg]]


 * From this graph we have that:
 * $$slope=0.000002\frac{m^2}{V}\,\!$$
 * We also have that the equation of slope is:
 * $$slope=\frac{2}{({7.8\times10^{-4}{I})}^{2}}\times\frac{m}{e}\,\!$$
 * Therefore we can calculate the ratio of $$\frac{e}{m}\,\!$$ by:
 * $$slope=0.000002\frac{m^2}{V}=\frac{2}{({7.8\times10^{-4}{I})}^{2}}\times\frac{m}{e}\,\!$$
 * $$\frac{e}{m}=\frac{2}{0.000002\times({7.8\times10^{-4}{I})}^{2}}\,\!$$
 * Where,
 * $$I=1.324 A\,\!$$
 * So we have:
 * $$\frac{e}{m}=\frac{2}{0.000002\times({7.8\times10^{-4}\times{1.324})}^{2}}\,\!$$
 * $$\simeq9.376\times10^{11}\frac{C}{kg}\,\!$$

Discussion on Error
Reasons for our systematic error
 * We had to measure the radius of the electron beam by eye using a fixed ruler in the back of the apparatus. This was hard because we had to roughly estimate for each of the measurements.
 * Looking at the second graph above, we can clearly see that there was a larger error for the ratio than the first graph.
 * Percent Error for the second graph
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$
 * $$\% error\approx4.33%\,\!$$
 * This percent error came out better than I thought it would be, so I assume that the error for the first graph would have been even smaller.

Acknowledgements

 * I would like to thank Ginny for the great help with this lab and all the other labs we have worked on.
 * I would like to thank Katie for helping us with the setup.
 * Alex Andrego and Anastasia Ierides for the great pictures and setup instructions.