# Physics307L:People/Wilkinson/E/MM

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# E/M Lab Summary

## History

Late in the 1800's, there was a hot debate concerning the nature of matter. The nature of subatomic particles was not well understood, indeed most scientists even believe that matter is continuous rather than partitioned into discrete particles. This all changed with experiments like those conducted by J.J. Thompson in 1897. Thompson was the first to determine the charge to mass ratio of the electron. This experiment was crucial in showing that matter is indeed partitioned into discreet particles. It also gives a ratio that can be used, with one other piece of information, to determine the charge or mass of the electron.

## Theory

We can start our discussion by mentioning the force a charged particle feels in a magnetic field.
$\displaystyle \mathbf{F} = q (\mathbf{v} \times \mathbf{B})$
This equation states that the charged particle will feel a force that is perpendicular to the velocity of the moving particle. Our setup will generate a magnetic field that will cause an electron particle beam to curve around on itself. The centripetal force of a particle moving in a curve is
$\displaystyle \mathbf{F} = \frac{m*{v^2}}{r}$
Combining these equations one can derive

$\displaystyle \frac{e}{m} = \frac{2 \times V}{B^{2} \times R^{2}}$
For our setup we know

$\displaystyle \mathbf{B} = (7.8 \times 10^{-4}) \times I$
Here we can prescribe the voltage, the current, and measure the radius of the resulting beam.

## Results

• Direct Calculation

We determined a value for the charge to mass ratio of the electron several different ways. First of all, the accepted value is (Steve Koch 01:09, 22 December 2010 (EST):Typo?)
$\displaystyle \frac{e}{m} = 1.76*{10^{11}} \frac{coul}{kg}$
The first calculation was calculated directly using the above equations.
$\displaystyle \frac{e}{m} = 1.43(6)*{10^{11}} \frac{coul}{kg}$
This calculation represents a 18% error and is 5 standard errors from the accepted value. This discrepancy represents some systematic error that will be further discussed below.

• Linear Regression 1

The next calculation uses google doc's spreadsheet (linest) program to calculate a linear relationship between the inverse current and the beam radius.

Inverse Current vs Radius

$\displaystyle \frac{e}{m} = 1.92*{10^{11}} \frac{coul}{kg}$
This calculation represents an 9% error. The error from the linear regression is too small to mention again representing some systematic error.

• Linear Regression 2

The final calculation uses google doc's spreadsheet program (linest) to calculate a linear relationship between the voltage and the square of the beam radius.

Voltage vs Radius^2

$\displaystyle \frac{e}{m} = 1.6(1)*{10^{11}} \frac{coul}{kg}$
This calculation also represents an 9% error. The error from the linear regression is 16 standard errors off the accepted value.

## Error

See Lab Notebook for sources of error.

## Acknowledgments

Thank you John Callow for ideas and an image.