Review Topics - Background q_2 E x_2 x_1 x_2 n da r q V q_1...

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Background x_2 x_1 x_2 - x_1 q_2 q_1 O Figure 1: Forces exerted by static charges. We develop Maxwell’s equations from their experimental foun- dations in differential and integral form for both static and dynamic applications. We consider first the microscopic and then the macro- scopic forms of the equations. Maxwell’s equations are numbered in what follows. Coulomb’s law Stationary point charges exert a mutual central force on one an- other. The force exerted on charge q 2 under the influence of q 1 in Figure 1 is governed by the following empirical law: F 21 = kq 1 q 2 x 2 - x 1 | x 2 - x 1 | 3 q 2 E 21 and is attributed to the electric field established by q 1 : E ( x ) = kq 1 x - x 1 | x - x 1 | 3 In MKS units, k = 1 4 π and = 8 . 854 × 10 - 12 F/m. The electric (Coulomb) force is a central force with a 1 /r 2 dependence on radial distance. The contributions from n point charges add linearly and exert the total force on the i th particle: F i = q i 4 π n X j 6 = i q j x i - x j | x i - x j | 3 E ( x i ) = 1 4 π n X j 6 = i q j x i - x j | x - x j | 3 lim q i 0 F i q i where we note that a point charge exerts no self force. For a con- tinuous charge distribution, the electric field becomes: E ( x ) = 1 4 π Z ρ ( x 0 ) x - x 0 | x - x 0 | 3 d 3 x 0 S V q E n r da Figure 2: Gaussian surface S enclosing a volume V with a point charge q within. where the primed and unprimed coordinates denote the source and observation points, respectively, and where ρ ( x ) is the volume charge density, with MKS units of Coulombs per cubic meter. Substitut- ing ρ ( x ) = n j =1 q j δ ( x - x j ) recovers the electric field due to n discrete, static point charges. Gauss’ law Evaluating Coulomb’s law for any but the simplest charge distribu- tions can be cumbersome and requires specification of the charge distribution a priori. An alternative and often more useful relation- ship between electric fields and charge distributions is provided by Gauss’ law. Gauss’ law will become part of a system of differential equations that permit the evaluation of the electric field even before the charge distribution is known under some circumstances. Consider the electric flux through the closed surface S sur- rounding the volume V containing the point charge q shown in Figure 2. According to Coulomb’s law, the flux through the differ- ential surface area element da is E · ˆ nda = q 4 π cos α r 2 da where r is the distance from the point charge to the surface area element, ˆ n is the unit normal of da , and α is the angle between ˆ n and E . Note next that cos αda is the projection of the area element da on the surface of a sphere of radius r concentric with the point charge. This projection could be expressed as ds = r 2 sin θdθdφ in spherical coordinates or, more generally, as ds = r 2 d Ω , where d Ω is the differential solid angle spanned by da as viewed from the point charge. Consequently, E · ˆ nda = q 4 π d Ω 1
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Integrating over the surface of S then yields (noting that the total solid angle subtended by any exterior closed surface is 4 π Sr.) I S E · ˆ nda
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