chap23 - ELECTROSTATIC ENERGY AND CAPACITORS 23 EXERCISES...

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23.1 ELECTROSTATIC ENERGY AND CAPACITORS 23 E XERCISES Section 23.1 Electrostatic Energy 14. Number the charges 50 C, 1, 2, 3, 4, i q i μ = = as they are spaced along the line at intervals. There are six pairs, so 2 cm a = 2 1 1 pairs 1 2 1 3 1 4 2 3 2 4 3 4 2 3 / ( / /2 /3 / /2 / ) ( / )(1 1 i j ij W kq q r k q q a q q a q q a q q a q q a q q a kq a = ∑ = + + + + + = + + + + 2 9 2 1 2 1) 13 /3 13(9 10 m/F)(50 C) /(3 2 cm) 4.88 kJ. kq a μ + = = × × = 15. I NTERPRET We find the electrostatic energy of a collection of point charges, using the method outlined in Section 23.1. D EVELOP The point charges are arranged on the corners of a square of side length a . Three of them are the same: The fourth has charge 1 2 3 . q q q q = = = + 1 4 2 . q = − q The first charge takes no work to bring into position, since there is no field present initially. 1 0 W = The work to bring in charge 2 takes the potential energy the charge will have due to charge 1: 1 2 2 q q W k a = The work to bring in charge 3 takes the potential energy due to the presence of charges 1 and 2: 1 3 2 3 3 2 q q q q W k k a a = + We’ve assumed that charges 2 and 3 are diagonally opposite on the square, thus the 2 term. The work to bring in charge 4 is the potential energy due to the presence of charges 1, 2, and 3: 3 4 1 4 2 4 4 2 q q q q q q W k k k a a a = + + Again, charges 1 and 4 are diagonally opposite each other, while charges 2 and 3 are separated from charge 4 by only the side length a . The energy of the configuration is the sum of these works. E VALUATE 2 3 1 4 1 2 3 4 1 2 1 3 2 4 3 4 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 q q q q k E W W W W q q q q q q q q a kq kq E a a = + + + = + + + + + = + + = + A SSESS This is not a particularly convenient method of calculating energies. Fortunately, the charge on a single electron is small enough that we can usually approximate real charge distributions as continuous and integrate. 16. I NTERPRET We find the electrostatic energy of a collection of point charges, using the method outlined in Section 23.1.
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