57. A narrow, negatively charged ring of radius R exerts a force on a positively charged particle P located at distance x above the center of the ring of magnitude kx F(x): (x2 + R2)3/2 where k > 0 is a constant (Figure 12). (a) Compute the third-degree Maclaurin polynomial for F. (b) Show that F -(k/R³)x to second order. This shows that when x is small, F(x) behaves like a restoring force similar to the force exerted by a spring. (c) Show that F(x)- -k/x? when x is large by showing that F(x) lim x00 -k/x2 Thus, F(x) behaves like an inverse square law, and the charged ring looks like a point charge from far away. F(x) х Nearly linear here х Nearly inverse square here FIGURE 12

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