25: Problem 25.57

INTRO:
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 1.80×10⁻¹⁸J 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.

Part A: 
The average distance an electron travels between collisions is 2.00 μm . What acceleration must an electron have to gain 1.80×10⁻¹⁸J of kinetic energy in this distance?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
a = 9.88×10¹⁷ m/s²

Part B:
What force must act on an electron to give it the acceleration found in part A?
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
F = 9.00×10^-13 N

Part C:
What strength electric field will exert this much force on an electron? This is the breakdown field strength. 
Note: The measured breakdown field strength is a little less than your calculated value because our model of the process is a bit too simple. Even so, your calculated value is close.
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
E = 5.63×10⁶ N/C

Part D:
Suppose a free electron in air is 0.700 cm away from a point charge. What minimum charge must this point charge have to cause a breakdown of the air and create a spark? Assume the electron is not hitting the charge particle but a neutral air atom that is still 2.00 μm away.
Express your answer to three significant figures and include the appropriate units.

SOLUTION:
q = 30.6 nC