Physics307L:People/Giannini/e/m

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My partner is Richard T. Meyers

Purpose

SJK 22:12, 13 October 2010 (EDT)
22:12, 13 October 2010 (EDT)
This summary is a bit off from what I'd like. As noted below, the final results are buried within the calculations, and the uncertainties are not next to them. It's worth talking with me in person about this next Monday.

To determine the charge to mass ratio of electrons using Helmholtz Coils (by applying different voltages and currents and then recording the new radii).

Calculating e/m (Overview)

SJK 21:45, 13 October 2010 (EDT)
21:45, 13 October 2010 (EDT)
It's tough to find your results buried in the calculations here. It would be better in your summary to present the three results concisely and put the calculations elsewhere. Also, I do not see a statistical comparison to the accepted value. You should say something like, "our measurement of 2.98 +/- 0.3 E11 C/kg is about 4 standard errors away from the accepted value of 1.76 E11 C/kg. So, it is very likely that we have significant systematic error in our measurement. This could come from a number of places ...

First we start by determining our Magnetic field, which changes with current.

$B=(7.793*10^-4weber/A*m)*I\,\!$

We then conbined three equations together:

$eV=\frac{mv^2}{2}\,\!$
$F_B=evB\,\!$
$F_c=mv^2/r\,\!$
• $F_B\!$ is the force of the magnetic field.
• $F_c\!$ is the centripetal force.

Solving for e/m:

$e/m=\frac{2V}{r^2*B^2}\,\!$

From this equation we solved for r^2

$r^2=\frac{2Vm}{(7.793*10^-4I)^2*e}\,\!$
• This equation is an equation of a line of r^2 verse V with a slope:
$slope=\frac{2m}{(7.793*10^-4I)^2*e}\,\!$

We then used LINEST in Google Docs to give us a linear fit of our data and got:

$slope=2.3(93)*10^{-5}+/-8.1(96)*10^{-6} m^2/V\,\!$SJK 21:38, 13 October 2010 (EDT)
21:38, 13 October 2010 (EDT)
This is not the correct way to report the uncertainty from LINEST. LINEST reports a slope of 2.393 * 10^-5 with an uncertainty of 8.196 * 10^-6. This means the slope is 2.393 +/- 0.8196 E-6. However, that's too many significant figures, so it should be reported as 2.4 +/- 0.8 E-6. Or I couldn't argue too much with 2.39 +/- 0.82, but most people would. To use the shorthand notation you tried out, it would be 2.4(8) E-6 or 2.39(82) E-6. When you put the number in parentheses, it means it's the uncertainty on the digit preceding it. Definitely ask me about this is if it doesn't make sense.

Using this result we calculated e/m.

$e/m=6.73(90)*10^{10}\,\!$

We also calculated e/m from a plot of r verse 1/I with constant V

$1/r=\sqrt{\frac{(7.793*10^-4)^2*e*I}{2Vm}}\,\!$

The linear fit line had a slope of: $slope=0.048(98)+/-0.0032(15)/m*A\,\!$ We have to invert this slope to get the correct one.

• We then calculated e/m again:
$e/m=2.98(45)*10^{11}\,\!$

The currently accepted value is:

$\frac{e}{m}=1.76\times10^{11}\frac{C}{kg}\,\!$

Conclusions

The r vs 1/I part of this experiment seems to yield better results then r^2 vs. V, in our case.SJK 21:46, 13 October 2010 (EDT)
21:46, 13 October 2010 (EDT)
"Better" based on what criteria? Looks nicer? Or has less systematic error?
Also, that we ended up with a value that is about 69% more then the currently expected value seems to show that this experiment has some sort of systematic error (since our measurements were representative of a r^2 vs V curve)SJK 21:47, 13 October 2010 (EDT)
21:47, 13 October 2010 (EDT)
This last sentence is close to what I said with my comment up above --- good! But you do need to go further and use the uncertainty in your measurement to decide if it's consistent. Not just the magnitude of the relative error.

Error(Experimental)

Possible sources of error: Human, Systematic: the equipment heating up over time, the lights being turned on and off during our experiment, the ruler we used being manufactured incorrectly, our old voltage sources may not supply constant voltage at all times.

Error(e/m)

SJK 21:49, 13 October 2010 (EDT)
21:49, 13 October 2010 (EDT)
I'm not sure if these are your correct uncertainties or not. In any case, you should report the uncertainties alongside the e/m measurements, to make it easy for the reader!
• r^2 Vs. V: $+/-3.3680*10^{10}C/kg\,\!$
• r Vs. 1/I: $+/-1.5394*10^{10}C/kg\,\!$

Citations

• Randy Lafler - Method for determining e/m