Chapter 20, Problem 73P

Figure 20.35 shows a thin, uniformly charged disk of radius R. Imagine the disk divided into rings of varying radii r, as suggested in the figure, (a) Show that the area of such a ring is very nearly 2πrdr. (b) If the disk carries surface charge density σ use the result of part (a) to write an expression for the charge d on an infinitesimal ring, (c) Use the result of (b) along with the result of Example 20.6 to write the infinitesimal electric field dE of this ring at a point on the disk axis, taken to be the positive x-axis. (d) Integrate over all such rings to show that the net electric field on the axis has magnitude

$E=2\pi k\sigma \left(1-\frac{x}{\sqrt{x^2+R^2}}\right)$

FIGURE 20.35 Problem 73