Electrostatics Setting:

Consider the discussion on this website:

As I am wont to do I was thinking about what to challenge you with for your portfolio for Electrostatics. There is a great deal of electrostatics from charge movement to forces to fields and electric potential embedded in the cathode ray tube which used to form the core of all TV's. I could ask you to explore how the cathode generates a beam of electrons or I could ask you to explain how the deflection plates work. What intrigues me the most is the electrostatic focusing device described partway down page linked at the top. I want you to focus your attention on the simpler form of the device (the second figure) that shows two tubes with some potential difference between them and a gap between the two tubes.

The statement that this is a focusing device suggests that electrons traveling parallel to the axis of the colinear tubes will be deflected towards the axis as they pass through the gap. Your ultimate task is to explain how this is reasonable given the principles of electrostatics that we have explored this term.

How are you supposed to do this you ask? Let me offer some suggestions:

I) Start by drawing a single long metal tube on which you place either net positive or net negative charge. Sketch the E field inside and outside the tube. Describe why any charge traveling inside the tube will feel no forces. What does this say about a "beam" of charges all traveling parallel to the axis of the tube? Why does it make sense that I can make this happen by connecting the tube to a "battery" which forces it to be at a + or - potential?

II) Now examine the illustration above and make sure you understand which tube has net positive charge and which has net negative charge. Make a larger more complete sketch of the equipotentials and electric fields shown in the figure above. In order to make sense of this you will need to be sure you understand why the equipotentials bend the way they do and where the E fields are stronger and weaker. There are some problems with the illustration that I want you to catch and/or clarify.

III) Explore the forces on several thoughtfully selected electrons traveling inside the tube and parallel to the axis of the tube. This involves sketching the trajectory of a charged particle through this field environment when the forces are changing direction and magnitude continuously. Does this quick conceptual analysis suggest that it might indeed be a focusing device? Explain your thinking clearly and completely.

I understand that this is not a trivial problem. Take your time and look for ways in which this problem is like problems you have done for skills validations and apply your concepts carefully. I am interested in how you use your developing conceptual knowledge in a new and challenging setting.