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By J. B. Tatum

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4 The equipotentials (V = constant) are straight lines in the uv-plane of the form v = fu. 7 x 2 + y 2 + 2 fy − 1 = 0. 8 Now aren't you glad that I chose f ? Those who are handy with conic sections (see Chapter 2 of Celestial Mechanics) will understand that the equipotentials in the xy-plane are circles of radii f 2 + 1 , whose centres are at (0 , ! 1 , 0). 7. The lines of force are the orthogonal trajectories to these, and are of the form x 2 + y 2 + 2 gy + 1 = 0. 9 These are circles of radii g 2 − 1 and have their centres at (0 , !

2 The D-field is ε0 times this, and since all the lines of force are above the metal plate, Gauss's theorem provides that the charge density is σ = D, and hence the charge density is σ = − Q. h . 3 10 σ = − This can also be written Q. 4 where ξ 2 = ρ2 + h 2 , with obvious geometric interpretation. Exercise: How much charge is there on the surface of the plate within an annulus bounded by radii ρ and ρ + dρ? Integrate this from zero to infinity to show that the total charge induced on the plate is −Q.

Well, if the uncharged body is metallic (as in the gold leaf electroscope), it is quite easy. In a metal, there are numerous free electrons, not attached to any particular atoms, and they are free to wander about inside the metal. If a metal is placed in an electric field, the free electrons are attracted to one end of the metal, leaving an excess of positive charge at the other end. Thus a dipole moment is induced. What about a nonmetal, which doesn’t have free electrons unattached to atoms? It may be that the individual molecules in the material have permanent dipole moments.

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