Chapter 22, Problem 68P

INTERPRET Ibis problem deals with the electric field of a charged disk. We want to show that the expre derived in Example 218 has the correct asymptotic behavior.

Develop From Example 22.8, the electric field on the axis (x > 0) of a charged disk of radius a is

$E_x=\frac{2kQ}{a^2}\left(1-\frac{x}{\sqrt{x^2+a^2}}\right)$

In the limit where x >> a, the quantity $x/\sqrt{x^2+a^2}$ may he approximated as

$\frac{x}{\sqrt{x^2+a^2}}={\left[1+{\left(a/x\right)}^2\right]}^{-}{}^{1/2}\approx 1-\frac{1}{2}\frac{a^2}{x^2}+\mathrm{....}$

EVALUATE Inserting the above approximation into the exact expression, we obtain

$E_x=\frac{2kQ}{a^2}\left(1-\frac{x}{\sqrt{x^2+a^2}}\right)\approx \frac{2kQ}{a^2}\left(1-1+\frac{1}{2}\frac{a^2}{x^2}\right)=\frac{kQ}{x^2}$

which is like that or a point charge.

ASSESS As always, a finite-size charge distribution looks like a point charge at a large distance.

68. Show that the result of Example 22.8 approaches the field of a point charge for x >> a. (Hint: You’ll need to apply the binomial approximation from Appendix A to the expression $1/\sqrt{x^2+a^2}$)