Chapter1.S07 - Chapter 1 Introduction and Survey 1.1...

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Chapter 1 Introduction and Survey 1.1 Maxwell’s equations in a vacuum 1.1.1 Electrostatics The results of the numerous investigations of electromagnetic phenomena carried out during the 18th and 19th centuries led to the development of a set of equations which govern these phenomena. Coulomb’s law (an action at a distance law) provided the description of the force, F 12 , on a stationary point particle at r 1 with an electric charge q 1 due to a stationary point particle with charge q 2 located at r 2 Note that the r 1 and r 2 are vectors. In the following equations of electrostatics the parameter k 1 is a constant determined by the system of units. In the modern system of units, SI, k 1 =(4 πε o ) 1 with ε o the permittivity of free space ( = 8.854 x 10 12 farad per meter (F/m) ). In Gaussian units , k 1 =1 See Table 1 on page 779. F 12 = k 1 q 1 q 2 | r 1 r 2 | 3 ( r 1 r 2 ) .... (1.01) where in SI units k 1 = 1 4 π± o =10 7 c 2 and q is in Coulombs (C) The concept of an electric f eld, E ( r 1 r 2 ) , generated by electrically charged point particles E ( r 1 r 2 )= k 1 q 2 | r 1 r 2 | 3 ( r 1 r 2 ) (1.02) allowed a local force law on point charges to be developed. F 12 = q 1 E ( r 1 r 2 ) (1.03) Superposition principle The superposition principle holds for the electric f eld and if { q i ,i , ..., N } are the charges of N point particles located at the points { r i , ..., N } the electric f eld at a point r 6 = r i is 2
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E ( r )= k 1 N X i =1 q i | r r i | 3 ( r r i 1 4 π± o N X i =1 q i | r r i | 3 ( r r i ) . SI units (1.04) The relationship between the charged particles and the f elds which they generate is generalized by Gauss’ law. This law relates electric f elds to electric charge densities, ρ ( r ) , in the area of electrostatics. · E ( r )=4 πk 1 ρ ( r ρ ( r ) o in SI units (1.05) Point charges and the Dirac Delta function In the case that ρ ( r (3) ( r r 2 ) ,where δ (3) ( r r 2 ) is a three dimensional ‘Dirac delta function’ (see next two pages) 1 , a solution to this partial differential equation is the electric f eld due to a charged point particle located at r 2 .The (one dimensional) Dirac delta function is de f ned by Z b a f ( x 0 ) δ ( x x 0 ) dx 0 = f ( x ) if a<x<b and 0 otherwise. (1.06) If x = a or x = b the integral is not well de f ned, but a common de f nition is Z b a f ( x 0 ) δ ( x x 0 ) dx 0 = 1 2 f ( x ) x = a or x = b The three dimensional Dirac delta function (recognized henceforth with vectors as variables) is equal to the product of three one dimensional Dirac delta functions δ ( r r 0 ) δ ( x x 0 ) δ ( y y 0 ) δ ( z z 0 ) (1.07) so that ZZZ V f ( r 0 ) δ ( r r 0 ) dx 0 dy 0 dz 0 = f ( r ) if r is interior to the volume, V and 0 otherwise (1.08) If the argument of a Diract delta function is a function of x, δ ( g ( x )) = X i δ ( x x i ) | dg dx | x = x i where g ( x i )=0 denote simple zeros of g ( x ) (1.09) 1 The Dirac delta function is de f ned by the condition that ] b a f ± x 0 ² δ ± x x 0 ² dx 0 = f ( x ) (1.1) if and ] b a f ± x 0 ² δ ± x x 0 ² dx 0 =0 (1.2) if x<a or x>b . If x = a or x = b the integral is not well de f ned.
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Chapter1.S07 - Chapter 1 Introduction and Survey 1.1...

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