q m of Electron Jeffrey Sharkey, Spring 2006

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1 of Electron Jeffrey Sharkey, Spring 2006 Phys. 2033: Quantu Lab 1 Purpose To observe and easure the eleentary ratio of electrons. 2 Methodology By controlling unifor agnetic field, we changed the orbital radius of electrons fired fro a cathode. By easuring the field reuired to hit several targets of a known radius, we can experientally calculate the ratio. 2.1 Euipent Used To observe the electron paths, we used an electron gun enclosed in a glass tube filled with Hg vapor. It was operated by applying one voltage to the cathode, and another voltage to the anode. We used a volteter to onitor the voltage applied to the anode, and an aeter to easure the current fro electrons captured by the anode. We also used two large Helholtz coils to generate and control a unifor agnetic field around the glass tube. Another volteter onitored the voltage we applied to the coils. 1

2 3 Collected Data Below is all data collected during the lab. Trial Voltage (V ) Current (A) Table 1: Measured applied voltage on Helholtz coils reuired to counteract natural agnetic field in roo. Current was calculated fro known relationship I = V. 0.1Ω 2

3 Radius () Voltage (V ) Current (A) Radius () Voltage (V ) Current (A) Table 2: Measured applied voltage on Helholtz coils reuired to bend electron bea to hit the target at the given radius. The left dataset has anode voltage V app = 22V, while for the right dataset V app = 28V. Shown are data fro three separate trials, each group separated by horizontal lines. Current was calculated fro known relationship I = V I 0.1Ω 0. 3

4 V acc Radius () I ± σ I I 0 ± σ I0 B(10 4 ) ± σ B (1011 ) ± σ / ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 3: For each anode voltage and radius cobination, the average current I and average counteracting I 0. Each σ is calculated fro the raw trial data shown in another table. Fro those values, the average B field and estiated ratio is calculated, along with their corresponding σ errors fro earlier fields. 4

5 4 Analysis and Results 4.1 Derivation of Electrons in a agnetic field can be accelerated by the force: F = v B (1) If the electron has an initial velocity perpendicular to the agnetic field, this force siply bends the path into a circular shape with radius r. This centripetal acceleration and radius can be described as: = vb (2) r Which can be rearranged to find our uantity : v2 = In the electron gun, the anode attracts electrons with a voltage V acc. When the electrons reach the anode (or siilarly pass through the slit in the anode), they have been accelerated to a kinetic energy: 1 2 v2 = V acc (4) Rearranging to solve for v, we can substitute into our earlier euation for : v Br (3) v = 2V acc (5) = 2V 1 acc Br We can suare both sides, and cancel out one set of the euation describing the ratio: [ ] 2 = [ 2V acc] (6) ters, giving us out final 1 (Br) 2 (7) = 2V acc (Br) 2 (8) We can control the anode voltage and agnetic field, and can easure the radius. Using easured values for each, we can then ake an estiate of the ratio for electrons. 4.2 Calculation of B and Thinking of a single charge a distance fro a coil, we can find the agnetic field B by integrating around the coil edge. For our solution, x is the distance fro the origin of the coil in a perpendicular direction. B = µ 0 2πR 2 I (9) 4π (R 2 + x 2 ) 3/2 5

6 When we cobine two coils together at a distance R identical to the coil radius, we create a very unifor agnetic field. The x ters in the cobined integration are canceled due to the syetry provided by the coils. This gives us: B = 8µ 0NI 125R (10) Actually calculating a value for B where I = 1.472A case, we know that our coil assebly has 72 turns and a radius of B = 8µ 072(1.472A) 125(0.33) = T (11) Finally, when bringing in our euation for radius is and V acc = 28V : fro above, for the case where our curl = 2(28V ) ( T ) 2 (0.0575) = C kg (12) We perfor these calculations after averaging the outcoes of several trials and show the results in a table above. 4.3 Earth s Magnetic Field Looking at out average value for I 0, we can calculate the natural agnetic field present in the lab. Fro our eight easureents, we found σ = , or an error of about 34.2%. B = 8µ 072(0.054A) 125(0.33) = T (13) The Earth s actual agnetic field is about T across ost of North Aerica. Our value was of the sae order, but 1 saller. We could explain this difference as experiental 3 error due to the difficult easureents, or we could infer that a positive agnetic field was soewhere above the apparatus, counteracting the Earth s field. 4.4 B Inside the Filaent Ideally, we want only an electric field in the gun. However, the filaent current introduces a sall B field between the cathode and anode. If no agnetic field were present, we would expect the electrons to spread out in a linear fashion. However, we observed the B field when we noticed that electrons coing fro the slit actually bent upward and downward; not following the expected linear odel. 6

7 5 Error Analysis In this lab, we perfored at least three trials of each easureent. This allowed us to calculate the standard deviation σ for each easureent, and to propagate these error estiations through the euations we used. For cobining errors, we used the euation introduced in an earlier lab: σ AB = σa 2 + σb 2 (14) However, in our case we also included the derivative of the euation being used: B = 8µ (0.33) = (15) = 2(22V ) ( ) 2 (0.0575) 2 = (16) For the averaged values of I, we calculated σ = 0.026, indicating an average error of 1.441%. Such a low error is very encouraging, indicating that our easureents were perfored in a very unifor environent. However, for our easureent of the average I 0, we found σ = 0.018, or a relatively high error of about 31.8%. We can attribute this high error to difficult easureents ade with the huan eye. Finally, we copare our average value of = C to the actual known value kg C. The error copared to this accepted value is 24.2%, which is very high. kg However, all of our easureents were very unifor, yielding an average cobined error of only 3.29% for the ratio. This suggests that any errors we encountered were systeatic, and not rando. 6 Conclusion Looking over the entire lab, it was very encouraging to find such unifor results. An error of only 1.441% on average for I, and 3.29% for our final values of. However, we found wildly varying values for the I 0 correcting current, giving it an average error of 31.8%. While the earth s agnetic field was present in the lab, it was still sall. In addition, the calibration was done by eye, introducing huan error. Overall, our value of = C was fairly close to the accepted value. While kg it was off by about 24.2%, it was of the sae order. Given that all our easureents involved a huan eye trying to detect the location of a di light, I was surprised that we ended up so close. 7