A narrow, negatively charged ring of radius R exerts a force on a positively charged

Chapter 8, Problem 57

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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 F (x) = kx (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/x2 when x is large by showing that lim x F (x) k/x2 = 1 Thus, F (x) behaves like an inverse square law, and the charged ring looks like a point charge from far away. x R F(x) Nearly linear here Nearly inverse square