Illustration 22.3: Monopole, Dipole, and Quadrupole

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The Coulomb force law predicts that the force of attraction (or repulsion) falls off as 1/r2 as the distance between two charges increases. But nature rarely provides us with point charges. Molecules, for example, consist of positive and negative charges bound together by nonclassical forces that can only be explained using quantum mechanics. But electrical forces are still present even if positive and negative charges are bound, and we can develop useful force laws that approximate common charge distributions. Restart.

This Illustration allows you to study the force between a movable test charge and orientations of one, two, and four fixed charges. The force between the test charge and a single point charge, known as a monopole, obeys the Coulomb force law. A system consisting of two closely spaced charges of opposite polarity is known as a dipole. Two dipoles placed next to each other form what is called a quadrupole. What can you say about the differences in the force vs. distance graph for the three cases? Does one or more of the plots show a force that decreases at some rate other than 1/r2? If so, why isn't this a violation of Coulomb's law? Why does the force drop off more quickly with the addition of more charges?

When you add up the forces due to several charges, the net force experienced by other charges may be different from 1/r2 depending on the orientation, magnitude, and sign of the charges. For the dipole, the separation between the positive charge and the test charge is almost the same as the separation between the negative charge and the test charge. If the separations were the same, the two charges would be on top of each other, and the net force on the test charge would be zero. But these separations are not quite identical. When we add up these forces, for the dipole we get a net force that goes like 1/r3 and for the quadrupole we get a net force that goes as 1/r4.

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