PROBLEM #8: DEFLECTION OF AN ELECTRON BEAM BY AN ELECTRIC FORCE You are investigating different methods of treating malignant skin tumors without surgery. One suggestion is to use high energy electrons from a particle accelerator. For this to work, it is necessary to control the path of the electrons so that they precisely hit the tumor and do not damage the nearby healthy tissue. A colleague suggests using an electric force to both control the energy of the electrons and to deflect them to hit the tumor. They contend that the easiest way to create this electric force is to separate the electric charges on two parallel conductive plates by applying a voltage difference across them. In that case, the electric force on an electron moving between those two plates depends on the charge separation and does not depend on where the electron is between the plates. This constant force on an electron passing between parallel plates is the charge of the electron times the voltage between the plates divided by the distance between the plates. You decide to use a cathode ray tube (CRT) to model the particle accelerator. In the CRT, electrons are freed from a wire at one end of an evacuated glass tube. Within this tube, the speed of these electrons changes when they go through two parallel plates in a direction perpendicular to the plates. These plates have a known voltage difference across them that is precisely controlled by a power supply. After leaving these accelerating plates, the electrons enter between two other plates, the deflection plates, that are parallel to the electron s motion. These plates change the electrons path from by an amount controlled by the voltage difference across them. The electrons then leave the deflection plates and travel through the vacuum until they hit a phosphorous screen. To check your understanding of how to control the electron beam of the CRT, you decide to calculate how the deflection of the electron beam on the CRT screen depends on voltage differences across both the accelerating plates and the deflection plates, the geometry of the plates, the voltage differences across the plates, the properties of the CRT, and the properties of the electron. Because you do not know the small speed of the electrons entering the accelerating plates, you assume that is negligible. Read Sternheim & Kane sections 16.1, 16.8. Review 2.5, 3.6, 6.1, 6.2, 6.8. EQUIPMENT You have a Cathode Ray Tube (CRT) to create an electron beam. There are two different power supplies used to operate it. The first, a Cenco power supply, should be used for the heating element that ejects the electrons from a wire at one end of the CRT and for the high voltage difference that is used across the accelerating plates. A different power supply that only goes up to 18V should be used for the voltage difference across the deflection plates of the CRT. You also have banana cables and a DMM. Read the section Cathode Ray Tube and Accessories in the Equipment section. If equipment is missing or broken, submit a problem report by sending an email to labhelp@physics.umn.edu. Include the room number and brief description of the problem.
WARM UP 1. Draw a picture of the important components of the CRT. Use the equipment section as your guide but only include one set of deflection plates. On the picture, draw the charge separation on the accelerating plates. Check that your charge separation will cause the electron to speed up as it goes from one plate to the other. Draw the direction of the electric force on the electron as it passes through the acceleration plates. 2. On your picture of the CRT, draw the charge separation of the deflection plates. Draw the electric force on an electron as it passes between these plates. 3. Note any regions in the CRT where there is no electric force on the electron. Assuming the only force on the electron is the electric force, draw the electron s path as it travels between where it is first emitted and where you can observe that it hits the screen when both the accelerating plates and the deflection plates have a voltage across them. Draw and label arrows representing the force on the electron in each region where it has an electric force exerted on it. Also draw an arrow representing the electron s velocity and acceleration for each region where there is an electric force on it and for regions where there is no electric force on it. As a reference also draw the path of the electron when there is no voltage difference across the deflection plates. The distance between where the non-deflected beam hits the CRT screen and where the deflected beam hits is the deflection. 4. Specify any forces on the electron that are negligible compared to the electric force and explain why. 5. Qualitatively compare the motion of the electron moving in the gap between the deflection plates and that of a ball that you throw horizontally. 6. Use conservation of energy to determine the equation that relates the speed of an electron leaving the accelerating plates to the voltage difference between the plates and the properties of the electron. Assume that the speed of the electron entering the space between the plates is negligible. 7. Use conservation of energy to show that the magnitude of the constant electric force on an electron moving in the gap between two parallel plates is F = edv/d, where e is the charge of the electron, DV is the voltage difference between the plates, and D is the distance between the plates. 8. Put a useful coordinate system on your picture. Assume that an electron enters the gap between the deflection plates going parallel to the surface of the plates and leaves the gap going at some angle to the surface. Based on the direction of the force on the electron moving between the plates, determine which component of velocity changes during this motion. Based on the electric force equation from (7), write down the equation for the electron s acceleration in that region as a function of
time, the properties of the electron, the distance between the plates, and the voltage difference across the plates. 9. Use the constant component of the velocity and the length of the deflection plates to determine the time that the electron spends in the gap between the deflection plates. 10. Use your drawing and kinematics to determine the position and direction of the electron as it leaves the deflection plates. Add this to your drawing. 11. Use your drawing and geometry to write down an equation giving the position of the electron when it hits the screen in terms of the properties of the electron, the properties of the plates, the voltage differences across each set of plates. Compare that position to the position of the electron when the deflecting plates are turned off. That comparison gives the deflection of the electron. 12. Complete your solution by using the actual numbers that describe your CRT. Refer to the distances shown on the diagram of the CRT in the equipment section. Does your solution make sense? If not, check your work for logic problems or algebra mistakes. PREDICTIONS Calculate the deflection of a cathode ray tube electron beam as a function of the voltage difference across its accelerating plates, the voltage difference across its deflection plates, the dimensions of the accelerating plates and deflection plates, the properties of the electron, and the location of the parts of the CRT. Use this equation to make a graph of the position of the electron as a function of the strength of the electric force on the electron between the plates. EXPLORATION WARNING: You will be working with equipment that generates large electric voltages. Improper use can cause painful burns. One unfortunate student in a past year had a hole burned through their finger from improper use of the lab equipment. To avoid danger, the power should be turned OFF and you should WAIT at least one minute before any wires are disconnected from or connected to the power supply. Never touch the conducting metal of any wire. Follow the directions in the equipment section for connecting the power supply to the CRT. Check to see that the connections from the power supply to the high voltage and the filament heater are correct before you switch on the power supply. The voltage difference between the cathode and anode should be in the range of 250V-500V. After a moment, you should see a spot that you can adjust with the knob labeled Focus. If your connections are correct and the spot still does not appear, inform your lab instructor.
Before you turn on the voltage difference across the deflection plates, determine the position of the CRT electron beam when it is not deflected. This position takes into account the effect of all of the other outside forces on the electron. Speculate on the cause of forces? Determine how each of these outside forces would affect the electron beam if you changed the position or orientation of the CRT. Try it out and from the behavior of the electron beam, determine which of these possible outside forces are negligible. If some force is not negligible, change the position or orientation of the CRT to reduce the effect of this force as much as possible. Now apply a voltage across one set of deflection plates, noting how the electron beam moves across the screen as the voltage is increased. Write down the range of voltages for which you can make a good measurement of the deflection. Repeat this procedure for the perpendicular set of deflection plates. Decide on which set of plates you will use for your measurement. Using the set of deflecting plates you have decided on, keep the deflecting voltage constant and vary the speed of the electrons entering the deflection plates using the voltage difference between the accelerating plates. Use this information to choose the best range of accelerating voltages and deflection voltages for your measurement. Devise a reliable way of measuring the position of the beam spot. Be sure you have established the zero deflection point of the beam spot. How many repetitions of a set of accelerating plate voltages and deflection plate voltages are needed to determine the uncertainty of your deflection measurements. Write down your measurement plan. MEASUREMENT Measure the position of the beam spot as you change the voltage difference applied to the deflection plates. Make sure you take enough measurements at each point to get a reliable measurement. Estimate the uncertainties in your measurements. Repeat this set of measurements for several different accelerating plate voltage differences. Be sure to record your measurements with the appropriate number of significant figures and estimate of its uncertainty. ANALYSIS Draw a graph of your prediction equation of the deflection of the electron beam as a function of the voltage applied to the deflection plates. Put your measurements on this graph. Repeat for all of the different voltage differences you used for the accelerating plates.
CONCLUSION How do your data points compare to the graph of your prediction? If they are different, explain why. How does the deflection of the electron beam vary with the applied deflection plate voltage? How does it vary with the applied accelerating plate voltage? State your results in the most general terms supported by your data.