You sometimes create a spark when you touch a doorknob

Chapter 22, Problem 22.97

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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. Most such collisions are elastic, so the electron collides, accelerates, collides, accelerates, and so on, gradually gaining speed. But if the electrons kinetic energy just before a collision is 2.0 * 10-18 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 FIGURE P22.61 shows, 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. a. The average distance between ionizing collisions is 2.0 mm. (The electrons mean free path is less than this, but most collisions are elastic collisions in which the electron bounces with no loss of energy.) What acceleration must an electron have to gain 2.0 * 10-18 J of kinetic energy in this distance? b. What force must act on an electron to give it the acceleration found in part a? 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. d. Suppose a free electron in air is 1.0 cm away from a point charge. What minimum charge is needed to cause a breakdown and create a spark as the electron moves toward the point charge?