# Deflection of an Electron Beam by a Magentic Field

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Deflection of an Electron Beam by an Electric Field Nicole N Lab Problem 1. 4 – February 3, 2011 Problem Statement: We were asked to test the design of an electron microscope to determine how a change in the electric field affects the position of the beam spot. The goal is to find out how different variables, such as charge of the deflection plates providing a vertical electric field and initial velocity of the electron beam will affect the amount of deflection the electron beam experiences.

We modeled the situation with a Cathode Ray Tube (CRT).

Due to the known equation for the deflection of the beam, we predicted that the higher the voltage, the greater the deflection. Prediction: To begin making predictions for this experiment, we first made a diagram of what exactly our situation is, and how we can calculate the information we are looking to find, which is how the different variables will affect the total deflection of the electron beam passing through a Cathode Ray Tube with deflection plates emitting a vertical electric field.

The CRT was placed in its holder at a 32. 0? angle.

To describe the total deflection, we first denoted that the change in the y-direction (coordinate system and labeled parts are indicated in the diagram below) as the electron passes through the electric field of the deflection plates is y1, and the deflection of the electron beam after it leaves the deflection plates until it hits the screen is y2. Therefore the total deflection in the y-direction = y1+ y2 .

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Also, L1 is the length of the region when the electron is passing between the deflection plates, and L2 is the distance from the end of the plates to the screen at the end of the CRT.

Please see a diagram of the experiment below with the different variables labeled: Next, we had to produce equations to determine the values of y1 and y2. The deflection of y1 is equal to: y1=12aT2 and using the equation Fe=qE=ma, we input qEm for the value of a: y1=12qEmT2 Also, we need a measurable value for T and E, so T=L1v0, and the value of E is equal to the charge of the plates, VPlates, divided by the distance between the plates,s : Vplatess. Therefore when all components are substituted, the equation of y1= 12VPlatess•mqL12v02.

Before we could begin to make a prediction, we also needed to figure the equation for the value of the initial velocity, v0, of the electron beam. To do that we calculated the total energy by setting the potential energy equal to the kinetic energy: PE = KE >qVACC= 12mv02, so v0=2qVACCm, where VACC = the voltage difference between the two deflection plates. Then, we worked out the equations necessary to calculate y2. First, y2 was set equal to the final velocity multiplied by T2 which is the amount of time it took the electron to go from the deflection plates to the screen: y2=a•T2, nd since time = distancevelocity, we can input the value of a (as computed previously) and T2 (in terms of distance and velocity): y2=qEmL1•L2v0 and v0=2qVACCm So, y2=qEmL1•L22qVACCm, resulting in q and m canceling out of the equation, leaving: y2=EL1•L22VACC Where again E = VPlatess > y2=VPlates•L1•L22sVACC. When y1 and y2 are added together, the total deflection = VPLATES2SVACC(L12+ L2) Experiment and Data: For all of the above equations, we have known values for the constants of s, q, m, L1 and L2: Constant| Value| s| 3. 0 x 10-3m| L1 | 2. x 10-2m| L2 | 7. 4 x 10-2m| So using the above equations and constants, we tested 7 different data points for various values of voltage with the charge difference between the plates (Vacc) at 543 V. Here is a table of the predicted values based on the charges. With Vacc = 543 V: Charge (VPlates)| y1 (m)| y2 (m)| Total Deflection = y (m)| 0 V (control)| 0| 0| 0| 1. 01 V| 6. 200E -5| 4. 590E -4| 5. 210E -4| 4. 64 V| 2. 848E -4| 2. 108E -3| 2. 393E -3| 8. 13 V| 4. 991E -4| 3. 693E -3| 4. 192E -3| 12. 53 V| 7. 692E -4| 5. 692E -3| 6. 461E -3| 16. 8 V| 1. 31E -3| 7. 632E -3| 8. 663E -3| 18. 4 V| 1. 130E -3| 8. 359E -3| 9. 488E -3| After setting up the CRT, we turned it on and attempted to arrange the screen so that the deflection would follow along with the x and y-axis’ respectively. Without any deflecting voltage from the plates, we took this point on the screen to represent our center, which was equal to the coordinates (-2. 5, 0). From here we measured the deflection for 7 different voltages with the cathodes connected for a potential difference at 500 V (actual voltage was measured at 543 V).

Below is a table of the values we received for the coordinates, along with a calculation of the difference between the coordinates and our origin: Voltage| CoordinatesAt 543 V| Deflection at 543 V| 0 V| (-2. 5, 0) origin| 0 mm| 1. 01 V| (-2. 5, 0)| 0 mm| 4. 64 V| (-2. 5, -1)| 1 mm| 8. 13 V| (-2. 5, -2)| 2 mm| 12. 53 V| (-2. 5, -3. 5)| 3. 5 mm| 16. 8 V| (-2. 5, -5)| 5 mm| 18. 4 V| (-2. 5, -5. 5)| 5. 5 mm| Results: The level of uncertainty for these values is equal to ± 0. 0005 m. (ex . 001 m ±0. 0005 m). In addition, the level of uncertainty for the DMM when the charge difference between the plates was tested is ±0. 005 V.

The level of error in this experiment, however, is greater due to the size of dot from the beam in comparison to the coordinate grid we used to determine the measure of deflection. The figures that are being used in this experiment are so minimal, that being able to measure to 0. 0005 m is not really sufficient. Below is a graph of the predicted values for the y deflection total versus the measured values : Conclusion: Based on the results of this experiment, I feel that our prediction was accurate that the greater the voltage of the deflection plates that the electron beam passes through, the greater the amount of deflection.

Although the values of our measured deflections appear to be quite different from our predicted deflection values on the graph above, as mentioned in the Results section, the amount of the difference is quite minimal when considering the level of uncertainty in this experiment. With that being said, the graph clearly indicates for both sets of values that the amount of deflection increases as the voltage of the plates deflecting the beam increases, thereby upholding our prediction.