User:Emran M. Qassem/Notebook/Physics 307L/2010/09/13

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=The Ratio e/m for Electrons=

Purpose

 * The purpose of this lab is to experimentally measure charge to mass ratio (e/m) for electrons and study the effects of electric and magnetic fields on the charged particle. For N coils carrying current I with radius R, the magnetic field along the symmetry axis x is given by:
 * $$B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$$


 * In the Helmholtz configuration, $$x = \frac{R}{2}\!$$, The permeability of free space $$\mu = 4*10^{-7} \frac{weber}{amp- meter} \!$$. In our apparatus $$N = 130\!$$ and $$R = 0.15 \!$$ meters, therefore
 * $$B = (7.8*10^{-4} \frac{weber}{amp- meter^2}) * I \!$$
 * By measuring V,I, and the radius of curvature r we can determine the ratio e/m.


 * The accepted value for the ratio e/m is $$1.76*10^{-11} \frac{coul}{kg} \!$$.


 * Ideas are described in detail in Professor Gold's e/m Ratio for Electrons.

Equipment

 * 6-9V Power supply rated at 2A (Gelman Deluxe Regulated Power Supply)
 * Power supply of 6.0V max rated at 1.5A (Soar DC Power Supply model PS 3630)
 * 150-300V power supply rated at 40mA (Hewlett Packard 6236B)
 * 3 Voltmeters
 * e/m Experimental Apparatus

Safety
Always connect equipment while it is off, so as not to get shocked. If voltage and current are high enough, serious injury or death can occur.

Setup

 * Connected 6-9 volt power supply to the Helmholtz jacks with banana cables.
 * Connected the ammeter in series to verify that the amps were within acceptable range.
 * Also used the ammeter to measure the data for the experiment.
 * Connected the 6.0 volt power supply to the heater jacks and set to 6.3 volts with banana cables.
 * Used a volt meter in parallel to verify the voltage didn't exceed 6.3 volts.
 * Connected the 150-300 volt power supply rated at 40mA to the electrode jacks of the electron gun with banana cables.
 * Used a volt meter in parallel to verify the voltage was in range.
 * Also used the volt meter to measure data for the experiment.

Procedure
After connecting everything in the setup, we went through the procedure defined in the lab manual in chapter 2
 * Shut off the lights as to see the beam better.
 * Set the switch on the panel to e/m posiion.
 * Turned on Heater supply for 2 minutes before proceeding to warm up the electron gun filament.
 * Turned on the 200 volt source, now we saw a violet hue beam (figure: Initial Beam 1/2).
 * We then cranked the current adjuster nob up until the beam formed a loop unto itself.
 * We had to rotate the bulb a little so that the beam actually came back around to itself without spiraling (figure: Circle Beam 1/2).
 * We proceeded to collect data by varying coil current and accelerating voltage while measuring the radius of curvature of the resulting electron beam.
 * We had to push the bulb to the left to get better measurements of the radius.
 * The radius was calculated by measuring from the left and from the right and averaging the two (see data section).
 * 10 different combinations were used, holding accelerating voltage constant and varying coil current, then holding coil current constant and varying voltage (see data section).

Data
Data was entered and graphs generated. Click on the spread sheet tabs to view the results. Curve fitting was used to calculate the line fit and slope using the LINEST spreadsheet function.

Calculations
To calculate e/m we first determined B, the magnetic field, by the equation:
 * $$B=\frac{\mu R^2NI}{(R^2+x^2)^{3/2}}\,\!$$
 * $$B=(7.793*10^{-4}\frac{weber}{A*m})*I\,\!$$

We used the fact that the potential energy of the electrons in the electron beam equals the kinetic energy
 * $$eV=\frac{mv^2}{2}\,\!$$

We then calculated $$v\,\!$$, the velocity of the electrons, by equating two equations:
 * $$F_B=evB\,\!$$


 * $$F_B\!$$ is the force of the magnetic field.
 * $$F_c=\frac{mv^2}{r}\,\!$$


 * $$F_c\!$$ is the centripetal force.
 * $$F_B=evB\,\!$$

Solving for $$v\,\!$$:
 * $$v=\frac{eBr}{m}\,\!$$

Substitute v into energy equation and simplifying:
 * $$e/m=\frac{2V}{r^2*B^2}\,\!$$

From the plot of $$r^2\,\!$$ vs. $$V\,\!$$ with constant $$I\,\!$$:
 * $$r^2=\frac{2Vm}{(7.793*10^{-4}I)^2*e}\,\!$$

This is a line with slope:
 * $$slope=\frac{2m}{(7.793*10^{-4}I)^2*e}\,\!$$

From the graph $$slope=7*10^{-6}\pm 3*10^{-7}\frac{m^2}{V}\,\!$$
 * Plugging this result into the slope equation:
 * $$e/m=2.07*10^{11}\pm 1*10^{10}\frac{C}{kg}\,\!$$

From the plot of 1/r verse I with constant V
 * $$1/r=\sqrt{\frac{(7.793*10^{-4})^2*e*I}{2Vm}}\,\!$$

This is a line with slope:
 * $$slope=\sqrt{\frac{(7.793*10^{-4})^2*e}{2Vm}}\,\!$$

From the graph $$slope=16.66(41)/m*A\,\!$$
 * Plugging this result into the slope equation:
 * $$e/m=1.83*10^{11}\pm 8.5*10^{9}\frac{C}{kg}\,\!$$

The currently accepted value is:
 * $$\frac{e}{m}=1.76\times10^{11}\frac{C}{kg}\,\!$$

Error
The manual explains that the greatest error is in the radius measurements. This is because the glass distorts the actual radius of the beam. The manual also explains the systematic error caused by the electrons as they don't achieve their theoretical velocity. This is because the accelerating voltage is not uniform, and the electrons loose some velocity do to collisions with the helium atoms in the glass. These errors lead to an experimental value higher than the accepted value, which is what was indicated in the lab manual, and is the case with our experiment as well. Our reported error is the error calculated due to a linear fit of the data points.

Qualitative Experiments

 * When we rotated the glass envelope it cause the beam to spiral. This can be explained by vectors as a portion of the component of the velocity of the electron beam now points in the direction of the magnetic field and therefore unaffected by the Lorentz force, which acts perpendicular to a particle's velocity.
 * Reversing the polarity of the coils caused the beam to deflect downward. The beam only deflecting down at to about 45 degrees.
 * We then switched on the deflection plates. We noticed that as the voltage increased, the beam grew at an angle above the horizontal. It grew slowly from 40 volts to about 100 volts. It then grew much faster from about 100 volts to 110 volts. After 110 volts the beam stayed the same length but decreased in angle towards the horizontal.
 * Reversing the polarity of the deflection plates caused the beam to point down initially and then approach the horizontal from below as voltage was increased. When the beam hit the deflection plate, we saw the beam spiral down the tube.

Acknowledgments/Citations

 * Katie for helping us get our ammeter setup correctly (in series).
 * Dr. Koch for helping us to understand the error and what's going on in the Qualitative Experiments
 * Randy my lab partner for doing the algebra as I did the excel spreadsheet.
 * Dr. Golds clear and informative lab manual.
 * Alex's lab book which gave us an idea of what we need to do.


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