You sometimes create a spark when you touch a doorknob after shuffling your feet

Question

You sometimes create a spark when you touch a doorknob after shuffling your feet on a carpet. Why? The air always has a few free electrons that have been kicked out of atoms by cosmic rays. If an electric field is present, a free electron is accelerated until it collides with an air molecule. It will transfer its kinetic energy to the molecule, then accelerate, then collide, then accelerate, collide, and so on. If the electron’s kinetic energy just before a collision is 2.0×1018 J or more, it has sufficient energy to kick an electron out of the molecule it hits. Where there was one free electron, now there are two! Each of these can then accelerate, hit a molecule, and kick out another electron. Then there will be four free electrons. In other words, as shows below, a sufficiently strong electric field causes a “chain reaction” of electron production. This is called a breakdown of the air. The current of moving electrons is what gives you the shock, and a spark is generated when the electrons recombine with the positive ions and give off excess energy as a burst of light.

Suppose a free electron in air is 1.3 cm away from a point charge. What minimum charge is needed to cause a breakdown and a spark as the electron moves toward the point charge?

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

Electron kinetic energy is k.e = 2.0*10^-18 J

and the distance is r = 1.3 cm = 0.013 m

we know that workd done = change in kinetic energy

that is W = Dk.e = F*Dx = m*a*Dx

Dx is 2*10^-6 m

from this the acceleration of the electron is a = Dk.e/(m*Dx)

the acceleration is a = (2*10^-18)/(9.1*10^-31*2*10^-6) m/s2 = 1.1*10^18 m/s2

and the electrostatic force is F= E*q = ma ==> E = ma/q = (9.11*10^-31*1.1*10^18)/(1.6*10^-19) N/C = 6263125 N/C

now the charge is q'

E = kq'/r^2

q' = E*r^2/k

q' = (6263125*0.013^2)/(9*10^9)

minimum charge is needed to cause a breakdown and a spark as the electron moves toward the point charge is q'= 1.1760*10^-7 C = 117.60 *10^-6 C

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