The ability to make a beam of electrons is quite important. It is, e.g., a key c

Question

The ability to make a beam of electrons is quite important. It is, e.g., a key component in making x-rays for clinical use in x-ray machines. The first step in the production of such a beam is to accelerate a single electron in a setup similar to the following.

An electron is accelerated in the uniform field E (E = 4.75×103 N/C) between two parallel plates shown in the figure. (The thickness of the plates themselves is negligible, despite how fat they might look in this picture.)

The electron starts from rest near the negative plate and passes through a tiny hole in the positive plate. In designing such a system, how far apart do you need to space the two plates such that the electron exits with a speed of 4.32×106 m/s? (As we will see later in this course, this means it has enough kinetic energy to generate light when the electron finally hits phosphors on a screen!)

Explanation / Answer

E = electric field = 4750 N/C

d = distance between the plates

V = Potential difference between the plates = Ed = 4750 d

m = mass of electron = 9.1 x 10-31 kg

q = charge on electron = 1.6 x 10-19 C

v = speed gained = 4.32 x 106 m/s

using conservation of energy

Electric Potential energy = Kinetic energy

q V = (0.5) m v2

(1.6 x 10-19 ) (4750 d) = (0.5) (9.1 x 10-31 ) (4.32 x 106 )2

d = 0.0112 m

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