Electromagnetic Field Theory: A Problem Solving Approach Part 24. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Energy Stored in a Dielectric Medium 205 where we recognize that each bracketed term is just the potential at the final position of each charge and includes contributions from all the other charges except the one located at the position where the potential is being evaluated W àtil i2 ï i3 Vs 3 Extending this result for any number N of already existing free point charges yields 1 N 4n l 4 The factor of arises because the potential of a point charge at the time it is brought in from infinity is less than the final potential when all the charges are assembled. b Binding Energy of a Crystal One major application of 4 is in computing the largest contribution to the binding energy of ionic crystals such as salt NaCl which is known as the Madelung electrostatic energy. We take a simple one-dimensional model of a crystal consisting of an infinitely long string of alternating polarity point charges q a distance a apart as in Figure 3-29. The average work necessary to bring a positive charge as shown in Figure 3-29 from infinity to its position on the line is obtained from 4 as 4 2 4irea L 11111 --1--I 2 3 4 5 6 5 The extra factor of 2 in the numerator is necessary because the string extends to infinity on each side. The infinite series is recognized as the Taylor series expansion of the logarithm 2 3 4 5 . _ . X X X X 6 9 q q q q q q q q Figure 3-29 A one-dimensional crystal with alternating polarity charges q a dis- tance a apart. 206 Polarization and Conduction where x 1 so that W -Z -ln2 7 4ireo This work is negative because the crystal pulls on the charge as it is brought in from infinity. This means that it would take positive work to remove the charge as it is bound to the crystal. A typical ion spacing is about 3 A 3 X IO-10 m so that if q is a single proton q x 10-19coul the binding energy is W x 10-19 joule. Since this number is so small it is usually more convenient to work with units of energy per unit electronic charge called electron volts ev which are
