329lect02 - 2 Static electric elds Coulombs and Gausss laws...

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2 Static electric fields — Coulomb’s and Gauss’s laws Static electric fields E ( r ) are produced by static (non-time-varying) distri- bution of charges and obey and the electrostatic laws shown in the margin where ρ ( r ) denotes the net charge density in 3D volume. Over the next few Laws of electrostatics: · E = ρ / o ∇ × E = 0 lectures we will find out how these laws emerge from Coulomb’s law . At the most elementary level, each stationary point charge (electron or proton) Q is surrounded by its radially directed electrostatic field E given by Coulomb’s law, and in the presence of multiple charges the field vectors of all the charges are added vectorially (linear superposition holds) to obtain a superposition field E . Coulomb’s law specifies the electric field of a stationary charge Q at the origin as E ( r ) = Q 4 π o r 2 ˆ r as a function of position vector r = ( x, y, z ) , where o 1 36 π × 10 9 F/m is a scaling constant known as permittivity of free space , r = | r | = x 2 + y 2 + z 2 is radial distance from the charge, and ˆ r = r r radial unit vector pointing away from the charge. r = | r | ˆ r Q q ˆ r Force exerted by Q on q: F = q E E = Q 4 π o | r | 2 ˆ r with electric field With multiple Q’s superpose multiple E’s x y z This Coulomb field E ( r ) will exert a force F = q E ( r ) on any 1
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stationary “test charge” q brought within distance r of Q (see figure in the margin). If qQ > 0 , force F is repulsive (directed along ˆ r ), if qQ < 0 it is at- tractive — like charges repel, unlike charges at- tract. The existence of a Coulomb field accompanying each charge carrier in its rest frame 1 is taken to be a fundamental property of charge carriers (established by measurements). When multiple static charges Q n are present in a region, the force on a stationary test charge q can be described as q E in terms of a superposition field E = n Q n 4 π o r 2 n ˆ r n written in terms of the magnitudes and directions of vectors r n pointing from each Q n to q . Equivalently, we can write q x y z r - r n Q n r n r Position vectors of charges are referenced with respect to a common origin O O E ( r ) = n Q n 4 π o | r - r n | 2 r - r n | r - r n | , where r and r n now denote the locations of q and Q n with re- spect to a common origin — this form is more convenient when static electric field E is to be calculated for an arbitrary location r (independent of the test charge notion).
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