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1 9/6/01 Coulomb Force Law and Superposition Charles A. Coulomb was a French military engineer who measured the forces between electric charges, reporting his results in 1785. He found that the electric force between two charged particles was proportional to the product of the charges, inversely proportional to the square of the distance between them, directed along the line joining them, and repulsive for like charges and attractive for unlike charges. Thus if charge q 1 is placed at the origin and q 2 is at the point Pr (, , ) θφ , then the force F 21 on q 2 due to q 1 is given by Coulomb’ Law, Fr 21 12 0 2 4 = qq r πε ± Newtons [N] , where 12 0 9 8.854187282 10 1 10 Farads per meter [F/m] 36 ε π ≈× is part of the constant of proportionality and is known as the permittivity of free space or of vacuum . Note the force on q 1 due to q 2 is simply FF 12 21 = − , i.e., the force has the same magnitude, but is in the opposite direction. This set of notes is related to the material in Sections 3.1-3.3 in the text by D.K. Cheng
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2 Forces between a pair of arbitrarily located point charges: To generalize the above result to the case when q 1 is not at the origin, we designate the locations of q 1 and q 2 by the position vectors r 11 1 1 = (,,) xyz and r 22 2 2 = , respectively. The distance between charges now becomes R =− rr 12 and the unit vector from q 1 to q 2 is replaced by ± / Rr r () 21 R so that Coulomb’s force law becomes FR 21 0 2 4 = qq R πε ± [N] . We will generally use this form for conciseness; occasionally, we’ll need to expand it out in rectangular coordinates, where it looks quite formidable: F xy z 21 2 1 2 1 2 1 02 1 2 2 232 4 = + + −+ qq x x y y z z xx yy zz [( ) ± ± ± ] [ ( )( ) ] / [N]. Superposition of forces due to a collection of point charges: Experiment also shows that the force on a charge due to several charges may be found by superposition , i.e., by adding (vectorially) the force on the charge due to each of the other charges acting separately. Thus, the force on a charge q located at point r due to charges q N ,,, , located at r ,, , N , respectively, is R R R =++ + = = R R R R N N N n n n n N 1 01 2 1 2 2 2 0 2 0 2 1 44 4 4 ±± ± ± , " [N]
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3 where R nn =− rr is the distance between charges q and q n , and ± / Rr r n R () i s the unit vector between them directed toward q . Forces involving more complex distributions of charge: The superposition idea is an extremely powerful one that we will use again and again. In this case, it allows us to easily determine the force due to more complex distributions of charge. For example, consider the region V with volume charge density ρ V (') r at the location of the vector position r ' , where the prime designates the so-called source point. A small differential volume element dV' there contains a differential amount of charge, dq dV V '' = which produces a differential contribution to the force on q( ) r at r as follows: 22 00 (' ) ' ' ˆˆ [N] , 44 V qd V qdq d RR πε == r FR R where now, R ' and ˆ ) / R r . Summing over all the differential elements yields the total force on q , which in this case becomes an integral, 2 0 ˆ ) ' [N] .
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