E/M. Hunter Layman Bridgewater College 1/16/2016

Transcription

1 E/M Hunter Layman Bridgewater College 1/16/016

2 Abstract The charge to mass ratio of an electron was observed in the experiment. This experiment involved the use of a PASCO scientific Model SE 9638 e/m Apparatus. This e/m apparatus was made up of an electron gun sealed in a vacuum tube, two Helmholtz coils, and a control panel. The electron gun shot out a beam of electrons that was curved into a circular path by a magnetic field produced by the Helmholtz coils. The circle path of the electron beam was visible and the radius was measured using a mirrored scale on the back of the apparatus. This Helmholtz coils had a current I and the electron gun had an accelerating voltage V both measured with RSR 01V86B Digital multimeters. The values of I, V, and the measured radius of the circle were used to calculate the charge to mass ratio (e/m). The accepted value for the charge to mass ratio is ( ± ) C/kg. 4 The experimental value for the charge to mass ratio was found to be ( 1.7 ± 0.5) C/kg, which is consistent with the accepted value. It was noted that the uncertainty in the experimental value is rather large.

3 I. Introduction The charge to mass ratio of an electron was initially determined by J.J. Thomson in In this experiment this relationship is determined, in a similar manner to J.J. Thomson, through the use of the PASCO Model SE 9638 e/m Apparatus. The PASCO Model SE 9638 e/m Apparatus is made up of an electron gun, Helmholtz coils, and a helium filled vacuum tube. A beam of electrons is launched with a known potential, thus the velocity is known. There are two Helmholtz coils around the electron gun that create a magnetic field at right angles to the electron gun. This magnetic field causes the electron beam to move in a circular path. Using the potential, the current in the Helmholtz coil, and the radius of the circular path of the electrons, the charge to mass ratio was calculated. This derivation is detailed in the Theory section of this paper.

4 II. Theory This e/m apparatus combines the use of an electron gun with Helmholtz coils. The electron gun fires a beam of electrons inside of a Helium filled vacuum tube. The electrons collide with the Helium atoms causing the electrons to become excited. From Layman 1 it is found that an electron that is excited jumps down energy levels causing a photon to be released. This photon has a wavelength of (hc)a 0( n n λ = ke init n final n init final), (1) where λ is wavelength, h = ev s is the Planck s constant, a nm which is the Bohr Radius, k = (N m )/C is the Coulomb s constant, e = C is the charge of an electron, n init is the initial energy level, and n final is the final energy level. The electron beam is visible, because the wavelength corresponds to a wavelength value inside of the visible spectrum. The Helmholtz coils are a set of coils that are used to create a magnetic field. This magnetic field causes the electron beam that is fired to move in a circular motion.the derivation for e/m is adapted from a derivation found in the PASCO Scientific e/m Apparatus lab manual. The magnetic force ( F m ) for a particle of charge q with known velocity v in a magnetic field (B) is F m = q v B. Since the electron beam is perpendicular to the magnetic field, the cross product of v B = v B and the equation becomes F m = e vb, () where e is the charge of an electron. Since the electron is moving in a circular motion, there is a centripetal force (F c ) that is equal to

5 where F c = m v e /r, (3) is the mass of an electron, and r is the radius of the circular path of the electron beam. m e The only force acting on the electrons is due the the magnetic force, thus, which can be F m = F c written as e vb = m v e /r or e/m e = v /Br. (4) The potential energy of the electron beam is equal to accelerating potential times the charge of the electron 3. This can be set equal to the kinetic energy because of energy conservation, yielding e V = 1/mev, where V is the accelerating potential. Solving for the velocity shows that v = ( ev /m e) 1/. (5) The derivation for the magnetic field of a Helmholtz coil can be directly derived from the Biot Savart law. The Biot Savart law is Figure 1. The Helmholtz coils derivation wit the differentiable units labeled. 5 B = μ 0 Idl r 4π r, (6)

6 where d l is the differentiable unit of length, r is the distance from the coil to the center of the coil a certain height z from the coil, and μ 0 = m kg/(sa ) is the permeability of free space in a vacuum. The d l and r are clearly labeled in Figure 1. The direction of the magnetic field d B is set up in a way that allows for simplification. For every value of d B, other than the magnetic field in the z direction, there is a complementary value that is equal but opposite. This leads to a cancellation of magnetic field only leaving the magnetic field in the z direction. Therefore the magnetic field B can actually be called B z. From Figure 1, it can be seen through the use of Trigonometry that B z = d B cos θ. Equation 5 can now be rewritten as B z = μ 0 Idl r osθ. 4π r c (7) The value of r can be written in terms of the radius of the circle R and the distance from circle z using the Pythagorean Theorem yielding Using the definition cos θ = adjacent hypotenuse r = z + R. (8), it is shown that c osθ = R/ z + R. (9) Since d l and r are shown to be perpendicular, d l r = d lr. By using this cross product result and Equations 8 and 7, Equation 6 can be rewritten as B z = μ 0 IR 4π (z +R ) d l. (10) 3/ Performing the line integral over dl gives a value of This causes the magnetic field to be πr, which is the circumference of a circle.

7 B z = μ 0 4π IR π. (z 3/ +R ) (11) By algebra this simplifies to μ B z = 0 IR (z +R ). 3/ (1) Since there are two sets of coils an equal distance away, Equation 11 is multiplied by a factor of which leads to a magnetic field of B z = μ IR 0. (13) 3/ (z +R ) This derivation is for two circles a distance of z apart and does not produce the full derivation for Helmholtz coils, since in the case of Helmholtz coils there are multiple circles layered (coils) that need to be taken into account. To solve this problem the number of coils, N, is multiplied into the Equation 1 showing that B z = Nμ0IR (z +R ). (14) 3/ For this specific experiment the radius of the Helmholtz coils is equal to the separation, thus R = a and z = a / with a being the radius of the Helmholtz coils. These values can then be plugged into Equation 14 to show that the magnetic field for the Helmholtz coils is B z = Nμ I 0 a(5/4) By plugging in Equations 15 and 5 into Equation 4 it is found that e/m V (5/4) a e = (Nμ Ir) 0. 3/ (15) 3. (16)

8 III. Experimental Procedure The lab manual provided is for the Model SE 9638 e/m Apparatus, while this experiment used the Model TG 13 e/m apparatus, which is set up in the same way. This experiment involved the use of an e/m tube, Helmholtz coils, a control panel, and a mirrored scale. Along with these pieces of equipment, a set of 3 power supplies and RSR 01V86B Digital Multimeters were needed for providing and measuring the voltage and current. The 3 power supplies were a 6 9 VDC for the Helmholtz coils, a 6.3 VDC for the heater, and a VDC that was used for the accelerating potential. The e/m tube,shown in Figure, was filled with Helium at a pressure of 10 mm. Hg, and also contained an electron gun and deflection plates. The electron beam left a visible light in the tube because the electrons collided with helium atoms, thus exciting the electrons and causing them to emit photons as described in the Theory section of the paper. The electron gun, shown in Figure, was comprised of a heater, an anode, a cathode, and a grid. The heater heated up the cathode which caused the cathode to emit electrons. These electrons were accelerated by a potential that was applied between the anode and the cathode, which was kept in focus by the grid. The Helmholtz coils were geometrically set up in a way that the the radius of the coils was equal to their separation. The control panel of the e/m apparatus was very straightforward and all the necessary hookups were provided in the PASCO e/m apparatus lab manual as shown in Figure 4. The mirrored scale was what allowed you to measure the radius of whatever circular electron configuration is produced. This mirrored scale was at the back of the apparatus shown in Figure 3.

9 Figure. The parts inside of the e/m vacuum tube. Figure 3. The parts of the e/m Apparatus.

10 Figure 4. The Control panel and where to plug in power sources and multimeters. First the toggle of the e/m Apparatus was flipped to the Measure position and the current adjust knob for the Helmholtz coil was turned to the all the way off position. The power supplies and the multimeters were hooked to the control panel in a way shown in Figure 4. The power supply for the heater was set to 6.3 V. The power supply for the accelerating voltage was set to be 00. ± V and the power supply for the Helmholtz coils was set to 7.8 V. Next the current adjust knob for the Helmholtz Coils was turned clockwise. Careful attention was paid to make sure that the current did not exceed Amperes, and was found to actually be ± A. After all the necessary voltages and currents were set, ten minutes elapsed in order to allow the heater to heat the cathode up. After the ten minutes the visible electron beam was seen to follow a circular path. The electron beam was then checked to be parallel with the Helmholtz coils. The radius of the electron beam was found by turning off the lights and measuring the

11 radius of the circular path from both sides. This value was then averaged to find the radius of the electron beam, which was r = ± m. Power Supply to Electron Gun (accelerating voltage) 00. ± V Helmholtz Coils Current ± A Power Supply to Electron Gun Heater Power Supply to Helmholtz Coils 6.3 V 7.8 V Radius of the Electron Beam ± m Table 1. This table shows the measurements found from the experiment.

12 IV. Data Analysis and Results The accepted value e/m is found by taking the known value of e = C and dividing it with m = kg, which yields a value of e /m = ( ± ) C/kg. Using Equation 16 and plugging in the values of I = A, V = 00. V, a = m, N = 130 turns, μ 0 = m kg/(sa ), and r = m, it was found that the experimental value of e/m is C/kg. In order to calculate the uncertainty in e/m the partial derivatives of Equation 16 need to taken with respect to V, a, I, and r. These partial derivatives are f V (5/4) a = (Nμ 0 Ir) 3, (17) f a 4V (5/4) = 3 a (Nμ 0 Ir), (18) f I 4V (5/4) a = (Nμ 0 r) I 3 3, (19) and f r 4V (5/4) a = (Nμ 0 I) r 3 3. (0) These partial derivatives are then used in the error propagation equation to show that δe/m e = ( f δ V ) + ( f δ V a) + ( f δ I) + ( f r) a I r δ. (1) To calculate the uncertainty in E/m the values δ V = V, δa = m, δ I = A, and δ r = m need to be plugged into Equation 1 along with the values used to calculate the experimental e/m value. This results in an uncertainty of C/kg. Therefore the experimental value for e/m is rewritten as ( 1.7 ± 0.5) C/kg. This means that the experimental value is consistent with the accepted value of e/m.

13 V. Conclusions This experiment has produced a value of e/m that is consistent with the accepted value. One concern when looking at the experimental value of e/m is the size of the uncertainty. An uncertainty of C/kg is very large. This takes away from the experiment and leads to less meaningful results. It is possible that the uncertainty in δ a was over approximated. The uncertainty in a was not provided in the lab manual and was therefore assumed to be the same as δ r. Since a is given in centimeters the approximation was half of a centimeter, which was the approximation made for measuring centimeters with a ruler. It is possible though that the uncertainty should have been lower than if the measurement was precisely made (this was not specified in the lab manual either). If the uncertainty in a should have been lower, this could have lead to an uncertainty in e/m that is smaller and more meaningful. If this experiment was to be reproduced in the future, more trials for the value of r would be performed at different values of I and V. This would lead to a better approximation of the radius and produce more meaningful results. The results of this future experiment could lead to a graph of accelerated voltage vs. r, whose slope could be used to determine the value of e/m. This slope could have been precisely calculated using Excel and might have lead to a value of e/m that has a lower uncertainty.

14 VI. References 1 Layman, H. (016). Spectrometry. Bridgewater College. Instruction Manual and Experimental Guide for the PASCO scientific Model SE 9638 e/m Apparatus. (n.d.). 3 Young, H., & Freedman, R. (01). Sears and Zemansky's university physics: With modern physics (13th ed., International ed.). Boston [Mass.: Addison Wesley. 4 (n.d.). Retrieved January 16, 016, from bin/cuu/value?esme search_for=atomnuc! 5 Griffiths, D. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, N.J.: Prentice Hall.