Chapter 21, Problem 93GP

(III) A thin ring-shaped object of radius a contains a total charge Q uniformly distributed over its length. The electric field at a point on its axis a distance x from its center is given in Example 21–9 as

$E=\frac{1}{4\pi {ϵ}_0}\frac{Qx}{{(x^2+a^2)}^{\frac{3}{2}}}.$

(a) Take the derivative to find where on the x axis (x > 0) E_x is a maximum. Assume Q = 6.00 μC and a = 10.0 cm. (b) Calculate the electric field for x = 0 to x = +12.0 cm in steps of 0.1 cm, and make a graph of the electric field. Does the maximum of the graph coincide with the maximum of the electric field you obtained analytically? Also, calculate and graph the electric field (c) due to the ring, and (d) due to a point charge Q = 6.00 μC at the center of the ring. Make a single graph, from x = 0 (or x = 1.0 cm) out to x = 50.0 cm in 1.0 cm steps, with two curves of the electric fields, and show that both fields converge at large distances from the center. (e) At what distance does the electric field of the ring differ from that of the point charge by 10%?