User:Randy Jay Lafler/Notebook/Physics 307L/2010/09/20

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e/m ratio

 * Purpose
 * The purpose of this experiment was to measure the ratio e/m.

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

 * Beware of shock

Set up
=Data=
 * We connected the 6-9V power supply to the Helmholtz jacks with BCN cables.
 * Connect an ammeter in series with the power supply and the jacks to measure the current
 * Next connect the 6.0V power supply to the heater jacks making sure not to exceed 6.3V.
 * We connected a voltmeter in series with the power supply and the heater jacks to make sure the voltage did not exceed 6.3V.
 * We connected the 150-300V power supply rated at 40mA to the electrode jacks of the electron gun.
 * This is the voltage that accelerates the electrons, and is the voltage we used in our equations.
 * We also connected a voltmeter in series witht the electrode jacks and the power supply to take better measurements.

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

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.
 * $$F_c=mv^2/r\,\!$$


 * $$F_c\!$$ 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 verse V with constant I:
 * $$r^2=\frac{2Vm}{(7.793*10^-4I)^2*e}\,\!$$

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

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

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:
 * $$e/m=1.83*10^{11}+/-8.5*10^{9}\frac{C}{kg}\,\!$$

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

=Error=

The manuel indicates that the greatest form of error is in the measurements of the radius. The reason is because the electron beam is enclosed in a glass envelope and the glass distorts the actual radius of the beam. Their is also systematic error in this experiment because the electrons don't percisely achieve their theoretical velocity. One reason is because the accelerating voltage is not uniform, and another reason is because the electrons loose some velocity do to collisions with the helium atoms in the glass envelope. Both of these errors leads an experimental value higher than the accepted value, which is the case with our experiment as well. The error we reported in our measurements is the error calculated do to a linear fit of our data points.

=Qualitative Experiments= =Acknowledgments= Professor Koch and Katie for helping us set up the experiment as well as helping us to find the proper power supplies.
 * When we rotated the glass envelope it cause the beam to spiral, because a component of the velocity of the electrons in the beam was now 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 bean stoped deflecting down at about 45.
 * We then switched on the deflection plates. Ve noticed that as the voltage increased the beam grew at an angle to the horizontal.  It grew slowly from 40V to about 100V, it grew much faster from about 100V to 110V, and after this point the beam stayed the same length but decreased in angle with the horizontal.
 * Reversing the polarity of the deflection plates caused the beam to point down at the start and to approach the horizontal from below as voltage is increased. When we had the beam hit the deflection plate we saw the beam spiral down the tube.


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