16[H]) You remember your physics professor mentioning that according to special

Question

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.

My Question

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.

Explanation / Answer

The Dielectric Breakdown Constant of air is 3*10^6 V/m.

lambda = h/(2*E*m)^0.5

E = h^2/2*m*(lambda)^2

E= (6.626*10^-34)^2/(2*9.11*10^-31*(2.43*10^-12)^2)

E= 4.13*10^-14

E/d = 3*10^6

d = (4.13/3) *10^-20 m = 1.38*10^-20 m

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