12[E]) You want to construct a simple particle accelerator and hit on the follow
ID: 1343406 • Letter: 1
Question
12[E]) You want to construct a simple particle accelerator and hit on the following design: you will take two oppositely charged plates with small holes drilled into them and set them up parallel to each other. You will create a vacuum between the two plates and fire a beam of electrons through the small hole in the negatively charged plate, aimed so that the beam will emerge through the hole in the positively charged plate. You want the electrons in the beam to gain 10.0 eV of energy, so you need to work out what magnitude of voltage difference to set up between the two plates.
Correct, computer gets: 1.0E+01 V
Hint: Recall that an electron-Volt (eV) is the energy needed to move an electron through a 1 Volt voltage difference
13[M]) Your electron beam consists of electrons that are initially moving at around 2.5×105 m/s. You want to work out how fast the electrons are moving after passing through your accelerator.
Correct, computer gets: 1.9e+06 m/s
14[E]) You decide to investigate the physical details of your electron accelerator a bit more closely. You start by determining the strength of the electric field between the plates. You've set the plates up so that their separation is 2 m.
Correct, computer gets: 5.0E+06 V/m
15[M]) The plates you chose had areas of 10 mm^2. You want to know how much electrical energy is stored by the electric field between the plates.
16[H]) You remember your physics professor mentioning that according to special relativity, mass and energy can be thought of as equivalent---that is the meaning of Einstein's famous equation. In some sense, the equation tells you that every kilogram of mass is equivalent to ninety million gigajoules of energy. You also remember your professor mentioning that according to quantum theory, particles actually tend to be smeared out a little in space---your professor specifically mentioned that (to oversimplify a bit) an electron can be thought of as having a wavelength of about 2.43×10-12 m. So you can think of the energy stored up in the electron's mass as spread out over a little cube whose side corresponds to the electron wavelength. You wonder how close you would have to bring the parallel plates of your accelerator so as to cause electrons and positrons (anti-electrons) to spontaneously pop out of the vacuum via conversion of the electric field's energy into electron/positron mass. You assume that you maintain the same voltage difference between the plates as they are brought closer together.
17[H]) You realize that the implication of your professor's comments is that the vacuum itself behaves like a dielectric medium and that it can undergo dielectric breakdown! This is due to the fact that a strong enough electric field in space will spontaneously create an electron/positron pair which is essentially the same thing as getting the vacuum to act as a conducting pathway for electrons. Naturally, you wonder what a plausible estimate of the dielectric breakdown strength of the vacuum would be.
Could someone please help me with numbers FIFTEEN (15), SIXTEEN (16), AND SEVENTEEN (17)?
Explanation / Answer
12) energy=10ev=10*1.6*10-19,voltage=energy/charge=10*1.6*10-19/1.6*10-19=10v
14)electricfield strength E=voltage/distance
E=10/2*10-6=5*106
15)capacityC=e0A/d=8.85*10-12*10-12/2*10-6=4.42*10-11 energy=1/2cv2=1/2*4.42*10-11*102=2.21*10-9joule
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