# 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

Total Apparatus
(bottom) Hewlett-Packard DC Power Supply (Model 6384A, 4-5.5V, 0-8A) and (top) BK PRECISION Digital Multimeter (Model 2831B, SER. NO. 000-03-0618 , 5 WATTS, 8 VA,50~60 Hz)
e/m Experimental Apparatus (Model TG-13)
(left of e/m E. A.) SOAR corporation DC Power Supply (Model 7403, 0-36V, 3A), (top right) BK PRECISION Digital Multimeter (Model 2831B, SER. NO. 099-10-0357 , 5 WATTS, 8 VA,50~60 Hz), (bottom right) Gelman Instrument Company Deluxe Regulated Power Supply (500 V, 100 mA)
measuring scale
image of electron beam
• 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

• The following equations were used to calculate the e/m ratio.
We found the Helmholtz configuration from Professor Gold's Manual to be:
${\displaystyle x=R/2\,\!}$, ${\displaystyle N=130\,\!}$, and ${\displaystyle R=0.15m\,\!}$
The permeability of free space is given as
${\displaystyle \mu =4\pi \times 10^{-7}{\frac {weber}{amp-meter}}\,\!}$
From these values we can calculate:
${\displaystyle B={\frac {\mu R^{2}NI}{(R^{2}+x^{2})^{3/2}}}\,\!}$
We know that the energy of the electron is equal to the kinetic energy:
${\displaystyle {e}{V}={\frac {1}{2}}{m}{v}^{2}\,\!}$
The magnetic force for a charge is...
${\displaystyle {F}_{B}={q}{v}{B}\,\!}$
The centripetal force is...
${\displaystyle {F}_{c}={m}{\frac {v^{2}}{r}}\,\!}$
Finally we set the centripetal force equal to the magnetic force and obtained:
${\displaystyle {\frac {e}{m}}={\frac {{2}{V}}{{r}^{2}}}{\frac {{(R^{2}+x^{2})}^{3}}{{({u}{R}^{2}{N}{I})}^{2}}}\,\!}$
According to Professor Gold's manual, the current accepted value of ${\displaystyle {\frac {e}{m}}\,\!}$ is:
${\displaystyle {\frac {e}{m}}=1.76\times 10^{11}{\frac {C}{kg}}\,\!}$

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

• Average
${\displaystyle {\frac {e}{m}}=1.54\times 10^{11}{\frac {C}{kg}}}$
• SEM
${\displaystyle {\frac {e}{m}}=2.62\times 10^{9}{\frac {C}{kg}}}$
• Percent Error
${\displaystyle \%error=12.4\%\,\!}$
The other way to find ${\displaystyle {\frac {e}{m}}\,\!}$ is to plot
1. ${\displaystyle r\,\!}$ vs. ${\displaystyle {I}^{-1}\,\!}$, where ${\displaystyle V\,\!}$ is constant.

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

From this graph we have that:
${\displaystyle slope=0.000002{\frac {m^{2}}{V}}\,\!}$
We also have that the equation of slope is:
${\displaystyle slope={\frac {2}{({7.8\times 10^{-4}{I})}^{2}}}\times {\frac {m}{e}}\,\!}$
Therefore we can calculate the ratio of ${\displaystyle {\frac {e}{m}}\,\!}$ by:
${\displaystyle slope=0.000002{\frac {m^{2}}{V}}={\frac {2}{({7.8\times 10^{-4}{I})}^{2}}}\times {\frac {m}{e}}\,\!}$
${\displaystyle {\frac {e}{m}}={\frac {2}{0.000002\times ({7.8\times 10^{-4}{I})}^{2}}}\,\!}$
Where,
${\displaystyle I=1.324A\,\!}$
So we have:
${\displaystyle {\frac {e}{m}}={\frac {2}{0.000002\times ({7.8\times 10^{-4}\times {1.324})}^{2}}}\,\!}$
${\displaystyle \simeq 9.376\times 10^{11}{\frac {C}{kg}}\,\!}$

## 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
${\displaystyle \%error={\frac {R_{accepted}-R_{measured}}{R_{accepted}}}}$
${\displaystyle \%error\approx 4.33\%\,\!}$
• 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.