Simple Nature - Light and Matter

the lightning strike.(b) Based on your answer from part a, which is more dangerous, alightning strike from a high-altitude cloud or a low-altitude one?(c) Make an order-of-magnitude estimate of the energy released bya typical lightning bolt, assuming reasonable values for its size andaltitude. E c is about 10 6 V/m.25 (a) Show that the energy in the electric field of a point chargeis infinite! Does the integral diverge at small distances, at large distances,or both? ⊲ Hint, p. 920(b) Now calculate the energy in the electric field of a uniformlycharged sphere with radius b. Based on the shell theorem, it canbe shown that the field for r > b is the same as for a point charge,while the field for r < b is kqr/b 3 . (Example 38 shows this using adifferent technique.)√Problem 26.Problem 27.Remark: The calculation in part a seems to show that infinite energy wouldbe required in order to create a charged, pointlike particle. However, there areprocesses that, for example, create electron-positron pairs, and these processesdon’t require infinite energy. According to Einstein’s famous equation E = mc 2 ,the energy required to create such a pair should only be 2mc 2 , which is finite.One way out of this difficulty is to assume that no particle is really pointlike, andthis is in fact the main motivation behind a speculative physical theory calledstring theory, which posits that charged particles are actually tiny loops, √ notpoints.26 The neuron in the figure has been drawn fairly short, butsome neurons in your spinal cord have tails (axons) up to a meterlong. The inner and outer surfaces of the membrane act as the“plates” of a capacitor. (The fact that it has been rolled up into acylinder has very little effect.) In order to function, the neuron mustcreate a voltage difference V between the inner and outer surfacesof the membrane. Let the membrane’s thickness, radius, and lengthbe t, r, and L. (a) Calculate the energy that must be stored inthe electric field for the neuron to do its job. (In real life, themembrane is made out of a substance called a dielectric, whoseelectrical properties increase the amount of energy that must bestored. For the sake of this analysis, ignore this fact.)⊲ Hint, p. 920√(b) An organism’s evolutionary fitness should be better if it needsless energy to operate its nervous system. Based on your answer topart a, what would you expect evolution to do to the dimensions tand r? What other constraints would keep these evolutionary trendsfrom going too far?27 The figure shows cross-sectional views of two cubical capacitors,and a cross-sectional view of the same two capacitors puttogether so that their interiors coincide. A capacitor with the platesclose together has a nearly uniform electric field between the plates,and almost zero field outside; these capacitors don’t have their platesvery close together compared to the dimensions of the plates, but638 Chapter 10 Fields