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# Opt1sol - 1 C\User\Teaching\505-506\F10\Opt1sol PHY 505...

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Unformatted text preview: 1 C:\User\Teaching\505-506\F10\Opt1sol.doc PHY 505 Classical Electrodynamics I Fall 2010 Optional Problems Set 1 with solutions Problem O.1. Two thin, straight parallel filaments, separated by distance d , carry equal and opposite uniformly distributed charges with linear density - see Fig. on the right. Find the electrostatic force (per unit length) of the Coulomb interaction between the wires. Compare the result with the Coulomb law for the force between the point charges, and interpret their difference. Solution : Using result (*) of the previous problem, we get d d E L d qE L F 2 2 ) ( ) ( . Note that the force drops with distance as 1/ d , rather than as 1/ r 2 for point charges. Such different scaling of interaction for systems of different dimensionality is very typical for physics at large. Problem O.2. Can one create the electrostatic fields presented below by sets of their components in Cartesian coordinates { x , y , z }, in a finite region of space? (i) { yz , xz , xy } (ii) { xy , xy , yz } Solution : Let us calculate curl of both supposed fields, using the definition of that operator - see, e.g., MA Eq. (8.5): y E x E x E z E z E y E x y z x y z , , E . For field (i) we get E = { x – x , y- y , z - z } 0, while for field (ii), E = { z – 0, 0 – 0, y- x }vanishes only in one point ( x = y = z = 0). However, according to Eq. (1.28) of the lecture notes, curl of real electric field has to equal zero at any point where it exists; hence field (i) can be created in a region of finite size, while field (ii) cannot. (The fact that field (i) has zero divergence, i.e. requires ( r ) 0 within the region of its existence, does not prevent it from being realistic, because it may be created by electric charges outside of that particular region.) Problem O.3. Calculate the force (per unit area) exerted on a conducting surface by an external electric field. Compare the result with the definition of the electric field given by Eq. (1.6) of the lecture notes, and comment....
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Opt1sol - 1 C\User\Teaching\505-506\F10\Opt1sol PHY 505...

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